I don't see anything better written, so here's a guide to the essentials of a culture victory. I start by summarizing the sources of culture and the mechanics of golden ages and great people. I proceed with a description of the tech path and cultural policy path that I recommend. I will explain how to use a single city or multiple cities without settling the debate on which is better. Then I will talk about religion choices. I have an overview of wonders "needed" and "possible" as that is the hardest call to make without knowing from experience. Finally, there's a bit about building Utopia and overflow.
Sources of Culture Explained
City-based culture
This is your bread and butter and breaks down into five types and a sixth kicker.
Terrain-based culture: Primarily, this means landmarks, which generate 6 before the Freedom finisher and 12 afterwards. Sometimes this is also a Great Prophet holy site or two (usually when capturing enemy great prophets), which provides 3 culture after the Piety finisher but does not get multiplied by the Freedom finisher (only the faith does). Finally, the terrain can be enhanced by your pantheon, by a monastery, or if you are lucky enough to get El Dorado. This ends up being the primary source of culture in a city by the late game, as eight landmarks (give or take) give you 96 culture from terrain.
Building-based culture: Aside from some unique buildings for certain civs, they are your Monument, Amphitheater, Opera House, Museum, and Broadcast Tower. These are your top priority to build in cities after getting basics like the granary, library, and university in order (and monument itself if not provided by Legalism). There are also the religious buildings, such as the Cathedral, in this category.
Artist-based culture: The artist specialist produces 3 culture per turn and should always be staffed immediately - no ifs, ands, or buts - as long as the city doesn't starve as a result. This is because it also generates the great artists that end up producing the awesome landmark culture mentioned above, but that takes time.
Wonder-based culture: Most of the wonders have at least +1 culture inherently, but even those that do not will get enhanced by Constitution, a Freedom policy, to have 2 culture on top of whatever they natively have. This means that every Wonder will add to your culture, some more than others. Also, the Great Mosque has an extra benefit of providing a free cultural building, which you should add to its value. (National Wonders don't benefit from Constitution or from Divine Inspiration.)
Policy-based culture: In addition to the extra 2 culture per wonder granted by Constitution, the Tradition opener grants 3 culture in the capital, the Liberty opener grants 1 culture per city, and the Military Tradition policy in Honor grants 2 culture in each city with a garrison. I will also mention here that the French UA grants 2 culture per city before Steam Power.
Multipliers: The multipliers you can have are 25% from owning the Sistine Chapel, 33% from having a broadcast tower in that city, 33% from having Reformation and having a wonder in that city, 50% from having the Hermitage in that city, and 20% from having Alhambra in that city. The Hermitage and Alhambra should always go in the capital in order to multiply the wonders and landmarks there. These multipliers are additive. However, the Golden Age multiplier is not additive with these and will rather multiply the result after applying the city multiplier. The total multipliers including Alhambra during a golden age make culture in the capital worth a little over 3x its stated value before multipliers (and nearly that even without Alhambra).
Puppet culture
This is exactly like city culture except that they will almost never run artists, they will put cultural buildings near the back of the list, they cannot have religious buildings bought in them, they will not have wonders not already there on conquest, and everything has a puppet modifier of -25% added in (basically negating the value of the Sistine Chapel, if you have that wonder). Still, they don't increase cultural policy costs, and obtaining a capital as a puppet that has a few wonders in it really doesn't hurt.
Cultural city state culture
This is a source of culture that, like puppets, will not increase your cultural policy costs. The value is doubled if you are an ally instead of just a friend, and it increases as you progress through the eras. It is multiplied by the golden age effect only.
Culture due to religion
This is from the World Church founder belief. As with city states, it can be a very potent way of increasing culture per turn without increasing your cultural policy costs. As with the city state culture, it is multiplied by the golden age effect only.
Culture due to happiness
With mandate of heaven, a policy in Piety, half of your excess happiness becomes culture. This effect persists in a golden age, making it effectively 60% of your excess happiness. While this is not the first priority for building culture per turn, it should always be considered. Trading for luxuries from other civilizations, allying mercantile city states, building happiness buildings, finding or exploiting natural wonders, taking certain policies, building happiness buildings, and building happiness wonders all have positive effect and can add to your culture per turn, without increasing cultural policy costs (as founding another city would).
Culture from kills
Killing barbarians will give any civilization with the Honor opener culture, and killing any unit will give the Aztecs culture. The culture is based on the combat strength of the unit killed and does not get multiplied in a golden age.
Culture due to a golden age
All other sources of culture are multiplied by 20% to give you the culture due to a golden age. Golden ages can be obtained by building the Taj Mahal, through the policies Representation and Reformation, through happiness, or by using a great artist.
A note on Golden Ages
The base duration of a golden age triggered by Taj Mahal, Representation, Reformation, or happiness is 10 turns. This is extended to 15 turns with Chichen Itza, 20 turns with the Freedom finisher, and again by another 5 turns if you are Darius.
The base duration of a golden age triggered by a great artist is 8 turns. This is extended to 12 turns with Chichen Itza, to 16 turns with the Freedom finisher, and again by another 4 turns if you are Darius.
Excess happiness does not contribute to culture until you have Mandate of Heaven. Excess happiness also does not accumulate to a golden age when you are in a golden age, which means that you will most likely have only 1 or 2 golden ages from happiness in a given game. If you think you can get two, then get plenty of happiness from the get-go. However, getting only one is also very typical, so it can pay to play around that fact. You can sell your luxuries at the start and delay triggering your first golden age from happiness a long time, when the cultural multiplier will have greater effect and when you may have already built the Chichen Itza to extend the effect. It is also sometimes possible both to delay the first golden age and to increase happiness dramatically for a second golden age from happiness. However, ideally, you will spend the last third or more of your game in a golden age, leaving no time to get more golden ages from happiness.
Never use a great artist to start a golden age before opening up Freedom. Instead, plant it. Save the "eternal golden age" aided by great artists either for burning through Freedom faster or even plant them all the way up until you have the Freedom Finisher, which extends golden age durations from all sources including great artists.
A note on Great People
You can get great artists from faith (purchasing great artists with the Freedom policy tree), from Wonders (with the Tower of Pisa and the Louvre), from the Liberty finisher, from the Maya UA, and from accumulating points in cities. It is important to understand how each of these interact with the "great person counter," which, when incremented, will increase the costs of future great people from points.
Faith - no effect to the counter. Aim to get 3 great artists from faith during post-Industrial times (1000, 1500, and 2500 faith).
Tower of Pisa and the Louvre - increment the counter. The Porcelain Tower does too, but Hagia Sophia does not.
Liberty finisher - increments the counter.
Maya UA - increments the counter unless it's an admiral or general.
Babylonian UA - the Great Scientist from writing doesn't increment the counter, making Babylon a viable civ choice for culture.
Accumulating points in cities - increments the counter. However, it is possible to time two or more great people to pop at the same time in one city. In this case, they will increment the counter multiple times, but they will be at the reduced point cost.
In a typical game, you will generate only great artists with possibly one to three great engineers. The first one or two great engineers might be spawned in the capital if you build enough wonders to cause the great engineer points from wonders to create them. Getting one great engineer from wonders is ideal in most cases, as it makes it possible to get Alhambra upon teching to Chivalry. Because getting two great engineers from wonders is rarely ideal, you may want to time the first great engineer to arrive at the 200 point mark instead of the 100 point mark, but that might not be enough to prevent a second spawn. If necessary, time a great artist to appear at the same time as the great engineer (by decommissioning artists) if that will allow the great engineer to spawn in time for rushing Alhambra. There are four situations in which the second great engineer from points can be put to good use. The first is for rushing the Porcelain Tower and catching the first wave of Research Agreements, hastening Industrial, if it will allow a smooth transition into Freedom. The second is for rushing Christo Redentor, if you can get there fast enough. And the third is for combining with a great engineer from liberty in order to rush the Sydney Opera House (not that this is likely for most players). The fourth is simply to ensure that a low-production city can get a wonder up in it for the Reformation bonus. If none of these situations occur and an extra great engineer is generated, planting it can speed up everything including Utopia (as it becomes +8 hammers with Freedom).
You do not need to run scientists after getting the key cultural techs, which typically means Radio. If you have enough food, however, you can run scientists any time before that, but only after staffing artists and/or if required for timing Industrial. Do not, however, allow this to spawn a great scientist if you are playing for a faster finish. The only exception is if this is required in order to make the timing for opening the Industrial era and the Freedom policy.
Tech Path
Start by getting pottery, writing, mining, and one or two luxury techs, not necessarily in that order. At Deity you may tech to Construction first, but below that you may prefer Currency first (if it's a Petra location) or Philosophy first (otherwise, for National College). This isn't anything particular to a cultural victory, because your priority in the first 50 turns is to set up a strong base (cities founded, beakers per turn and the rest up and running) from which to proceed. Indeed one of the worst mistakes you can make is to think that early game teching is unimportant to a cultural victory, as it is paramount and, very early on, your beakers are even more important than your culture. Getting National College up in the 70s is a good idea.
After Philosophy (National College and the Oracle) and Construction (Terracotta Army), tech to Civil Service (Chichen Itza and food). After Civil Service, tech to Education (universities). After Education, if you are able to get the wonder, tech to Chivalry for Alhambra (see below regarding this). Otherwise, tech to Acoustics for Sistine Chapel and the Opera House. After Acoustics, tech to Metal Casting and buy the Workshop in the capital. After Metal Casting, if you are able to get the wonder (easy on emperor or below), tech to Printing Press for the Leaning Tower of Pisa. Otherwise, tech to Architecture (Hermitage and Taj Mahal). After Architecture, tech to Navigation. Use Oxford to unlock Archaeology (Industrial era, museums, the Louvre). After Archaeology, tech to Scientific Theory (Public Schools). After Scientific Theory, tech to Radio (broadcast towers, Eiffel Tower). After Radio, tech to Fertilizer and Industrialization (food, production).
At this point you will either have a very strong level of science (possible with 4-5 cities and high population and observatories) or an anemic one (likely with 1-3 cities and no observatories). If you are in the strong science boat, tech to Plastics for Christo Redentor (use a GE - the Liberty finisher can provide it). If you have at least moderate science, get the Statue of Liberty with Replaceable Parts for a faster Utopia and tech to Plastics just to use the research lab or Christo Redentor for overflow when building Utopia. Otherwise, tech to Railroad, Biology, and Flight for some more production and to prepare a defense.
Multiple City Cultural Path
The most important policies for a cultural victory are in Freedom, but you can't get them until the Industrial era. What you will need to get before then are the policies Representation (in Liberty) and Religious Tolerance (in Piety) in order to reduce the costs of policies. You are not, however, necessarily in a hurry to fill out piety. This will increase your culture per turn, but it will do nothing for your beakers per turn, causing you to get more policies before Industrial than you would otherwise and slowing down your completion of Freedom. What you need to start with are policies that will improve your science.
Those policies are Landed Elite and the Tradition Finisher (for more food), a policy tree that also gives you the much-needed Aristocracy (for wonder production) and Monarchy (for happiness). So you should start with filling out full Tradition. The faster you finish the Tradition policies, the more benefit it will provide to your science (and production and gold).
So your order of policies is full Tradition, Liberty Opener, Citizenship, Representation, Piety Opener, Mandate of Heaven, Organized Religion, Reformation, and then Religious Tolerance. You do have the option of mixing the order of the preceding policies, but you want exactly those policies before getting to Industrial. Then you want the Freedom Opener, Constitution, Civil Society, Democracy, Free Speech, and Universal Suffrage.
Freedom provides more great people production, less food required to feed the artist specialists, longer golden ages, and twice as much culture from landmarks. You will be painting the tiles around your capital with 8 or more landmarks and using the other great artists to start golden ages, so this is all critical to your strategy. You will win in less than 80 turns of completing the Freedom tree, conservatively, and that is with the ten most expensive policies ahead of you and building Utopia.
The last policy tree can be Patronage, Honor, or Commerce. Patronage is usually the best bet because of the science from Scholasticism, but Honor and Commerce have the potential to provide more happiness and/or culture, if that can matter more in the few turns left. Go for Aesthetics and Scholasticism immediately after Freedom for the most benefit.
Completing Liberty will give you a great person. It should be used for a great artist unless you are able to get Christo Redentor with the GE and you finished Liberty immediately after Freedom instead of another policy tree.
More creative and talented players than I, who don't need this guide, may be able to forge a better path. This strategy works, however, and will reliably get you done in less than 300 turns even with the worst start and settings.
Single City Cultural Path
After filling out Tradition, immediately open Piety and take Mandate of Heaven, Organized Religion, Reformation, and Religious Tolerance. You're going to have roughly 2-4 policies to fill up besides full Tradition and Religious Tolerance before you get to the Industrial era, and you don't need Representation. I suggest going for Aesthetics if you are strong on science but relatively weak on culture, as that will make snapping up cultural CS easier. I suggest going for Scholasticism if you are weak on science, as you are wont to be with one city, because it will give you a much-needed boost. Do one or the other first, then take more policies in Patronage only if you don't make it to Industrial in time. You should still fill out Liberty after Freedom for the golden age and the great artist.
Map Scaling and Wide Strategies
On a standard map, additional cities increase policy costs (linearly, not exponentially) by a 15% factor that is reduced to 10% with Representation. On a large map, this goes down to a 10% factor reduced to 6.7% with Representation. On a huge map, this goes down to 7.5% reduced to a 5% factor with Representation. On a huge map, additional cities do not slow down your policy acquisition if they can get all their buildings up promptly (providing about 68 culture per turn in a Golden Age with the Sistine Chapel, Broadcast Tower, and five artists staffed) because you are not likely to achieve more than 1300 culture per turn with the capital alone. On the other hand, going very wide is completely ruled out as optimal, on a standard size map, where the break-even point is higher, because the scaling from great artist points degrades to the point where you don't get any more from founding more cities and because you have to start the city early enough to build a wonder and cultural buildings. It is, however, possible, if you are a glutton for pain!
How to Start Building
Your starting location should always be high in food and should never be coastal unless there are only four or five water tiles. If your start doesn't match that description, try to move onto a river and/or inland. Working landmarks and artists really taxes a city's growth without a strong food location. A mountain for the observatory is not as important as food, and it isn't needed for a typical 2-3 city strategy, but it is required for a 4-5 city high-science strategy and is highly desirable for a one-city game.
Unless you are on weird settings (tiny islands or no huts or Ethiopia), you will want scout first. You will want to get a free amphitheater in the capital, so build a monument before Legalism if you do not have a Petra start. If you do, either skip it or build Petra before researching Drama and Poetry. Skipping the monument entirely slows down finishing tradition, and that just isn't necessary for a cultural victory in the way it helps a science victory; indeed, it slows down your science.
After scout and monument, if you are going for a multi-city game, you will want a worker and possibly a granary and/or shrine before you start building settlers. You should not build anything else before getting your settlers sorted out (through gold or production). So much depends on difficulty level, start, and how much gold you can abuse out of the AI. But the earlier you start your additional cities, the better. They can justify their drag on the culture costs better if they can get more buildings up in them. They have a much better chance of hard-building a wonder if they get going fast. Also, after Legalism is unlocked, adding cities to your empire will accelerate very early policy acquisition in Tradition because the free monument will increase culture per turn by more than 15%.
Overall, two or three cities should be considered the bog standard. More than that are not necessary to push up your gold and science, unless you are going for technologies more advanced than Radio. And a single city requires you to pull out some tricks with puppets, interfaith dialogue, and/or scholasticism and/or have a good start in order to get the required science. The second and third city will provide roughly 120 culture per turn between the amount in the city, if you get a wonder and broadcast tower, and just a single planted great artist, which is better than the 10% increased cultural costs. If the game goes on long enough, they also provide a second great artist to extend the golden age.
If you are going for a single-city game, you really want a high-food location with a mountain and a river in order to have adequate science. The tiles in the city radius become the only tiles in the game, so they need to be good. If it isn't too great, then you will certainly do better with a multi-city game. If you can get science and gold by other means (including puppets) and aggressively secure cultural city states, you can finish very competitive times, possibly the fastest, with an OCC style. OCC without puppets, however, is not likely to be faster than one city with puppets or a strong 2 or 3 city game.
If you are going for a four or five city game, you really need to justify the investment with something special that you wouldn't get by having only two or three cities. The only thing that really fits that description is Christo Redentor and/or the Sydney Opera House. You really need to be aggressive on building both culture and beakers if you are going to get them, with high food and mountainside locations for all of your cities. Christo Redentor is much more likely than the Sydney Opera House, which will require an extraordinarily talented player and 2 great engineers (in a non-capital coastal city) to pull off in time to make the victory faster.
The basic priority for building order, outside of the capital and apart from wonders, is to have monument, library, granary, possibly a water mill or lighthouse or garden, university, workshop, amphitheater, opera house, possibly an observatory, museum, public school, and broadcast tower, in roughly that order. Shrines and temples are best built after opening piety whenever there is an opportunity. Happiness buildings are the next in priority (unless needed to avoid going negative). The wonder interrupts this order and needs to be up ASAP after getting the tech for it. Usually at least the university and/or workshop and/or opera house have to bought in order to get it all done reasonably fast.
The garden is very important only in the capital, where the wonders increase the base great artist point generation. The National College, Hermitage, the Ironworks, and Oxford University will be in the capital every game. You will also want to get the Great Temple when possible and the National Monument if ever there is a spare moment. The National Treasury is handy but, in a sort of catch-22, requires you to be able to afford marketplaces comfortably first; it can sometimes show up after broadcast towers in a multi-city game. With just 1 city, however, it is a high priority for picking up some of the gold not available through trade routes.
Whatever your number of cities, you will need to make it a priority to get cultural city state allies, and any city state allies are welcome. Scout early, scout well, and get some military units out to complete the barbarian camp quests before the AI gets to them.
Religion Choices
The strongest pantheon choices are desert folklore, desert folklore, desert folklore, stone circles, religious idols, goddess of festivals, and dance of the aurora. They will get you a religion out of a single shrine and generate the faith you need to get religious buildings, missionaries, and 3 great artists from faith. Without a faith pantheon, you need to invest in more buildings to get all this going.
However, there is an alternative, which your start may dish out to you. Taking a culture-generating pantheon (or even God of Craftsmen) can be combined with building Stonehenge or Hagia Sophia, buying off a religious city state, or settling a city next to a religious wonder such as Uluru, Sri Pada, Mt. Fuji, or Mount Sinai. Taking the divine inspiration follower belief will grant +2 faith per world wonder. On lower difficulties, with lots of wonders, this can generate more faith in any case.
For founder beliefs, you have five good choices.
Tithe or Church Property - more money.
World Church - more culture (better in the mid-level difficulties where other religions are weak but cities are large enough).
Pilgrimage - more faith for great artists (another idea if you have no good pantheon, also reliant on other religions being weaker as is World Church).
Interfaith Dialogue - more science (best on Emperor, Immortal, or Deity where other religions are strong).
Interfaith Dialogue takes some practice to be abused to great effect. You need a faith pantheon and high faith per turn to make it work. If you can enhance your religion in the medieval era and have about 50 faith per turn, it is great. Try to get the Great Mosque in order to improve the effect by 50% (3 use missionaries). Make sure you get open borders with a nearby religious civ with a large capital, but not with another civ. Run your missionaries into the other civ in order to "pre-attrition" them down to 250 strength. Then bomb the capital repeatedly. In one game you can get 6 missionaries, 3 times each, for 120 to 180 science each. That can be over a dozen turns of science. Stop buying missionaries in time to save for great artists in Industrial.
For follower beliefs, you have one mandatory choice, which is Cathedrals. On higher difficulties it is worth locking it in first, but on lower difficulties with a faith pantheon for fast religion you can wait until you enhance your religion. The reason it is so good is that it gives you an extra artist slot to use to generate more great artists.
For the second follower belief, you have three good choices.
Religious Community - possibly the best, as it can shorten Utopia and benefits all cities to get buildings and wonders faster.
Divine Inspiration - necessary if you do not get a faith pantheon.
Religious Art - The fact that it is in the "cultural capital" means that it benefits from the multipliers there to become roughly 25 culture per turn late-game.
If you are beaten to all of these, any of the other building choices (mosques, monasteries, pagodas) or Choral Music provides a bit more culture (the latter being preferable if you can't buy two rounds of buildings before Industrial or if you need to use the faith to buy missionaries or great prophets - i.e., if you haven't chosen tithe).
For the enhancer belief, religious texts or itinerant preachers can provide some oomph to tithe, church property, world church, or pilgrimage, while reliquary for extra faith is always good.
