First I'll go over resources generally, then talk about each victory type.
-No matter what victory type you're going for,
food is the most important resource at the beginning of the game. Without food your cities won't grow, ad without more citizens, all of your resources will be low.
-Whether or not
food remains important depends on your city's location, the state of the game, your civilization, and the victory you're going for.
Production is very important, right along with food. Depending on circumstances, one or the other may be more important. More on that later.
-Generally speaking,
culture and faith are the least important resources. Culture CAN be useful when going for a cultural victory, but it's more important to prevent other people from getting cultural victory. Only certain civs, with unique tile improvements focused on culture (such as Brazil) should worry about it, even if going for a cultural victory. You should only really focus on culture very late game if you're seriously worried someone else will win a cultural victory. Faith is even MORE situational. It's also only important when going for cultural victory, and mainly for civs focused on faith, such as Ethiopia.
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Gold is somewhere in the middle. It's more important for Diplomacy victories, but not until late in the mid-game. It's never very important early on, and will come in one way or another easily enough. It's useful and nice to have at times, but it will very rarely be more important than food or production.
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Science is almost always the single most important resource.
Now for victory!
-For
Science victories, the most important main resource will always be food. Science, if available, will generally trump food, even at the beginning of the game when your population is low. This is what makes Babylon so strong, for any victory type really, but especially Science. Faith remains nearly useless for science victories. Production is just as important as any victory; without production to build those science buildings, your science will falter. Later in the game, production becomes the single most important factor, because you need to build all the spaceship parts. Because of this, keep production in mind when deciding where to put cities and what to build in them. Production is also important to build more advanced military units than your opponents (since you are further in science than them, right?) Gold is in the middle, and culture is under that.
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Diplomacy victories are the one I have the most experience with, and they are quite unique. In the early game, food is the most important, but as you enter the mid-game, production becomes more important, and then this switches to gold towards the end of the mid-game. You need food to get the cities growing, just like any other victory type, but then you need production to start building them up. You are in it for the long haul, but science is less important than in science victories AND you need gold. This means you can take your time learning techs, but you need to build more buildings. Science doesn't need to focus on gold, but you do; the production buildings that science needs come later, but your gold buildings are in the mid-game. So, essentially, you're building more buildings but with less techs, thus production trumps food. Then, in the late game, you need gold, because you need to pay off them city-states to get them to be your allies! However, just because you're not going for science victory, don't think that science isn't important; it still is. Focus it as much as you can. Faith and culture, on the other hand -- pretty useless to you.
The other 2 victory types depend on which civilization you're playing, but the simplest is culture, so we'll discuss it first.
Culture victories can be won in 2 different ways:
1.
A few huge cities with lots of work of art slots and Great Artist/Musician/Writer generation. For the first victory path, food will remain the most important resource throughout the WHOLE game, period (though science remains very important). Culture itself is still not the most important resource, but GPP is (Great Person Points). Because of this, make sure you focus on all the specialists you can, even if you don't need the culture. Production and gold are both resources that are going to be less important to this victory type than for others. Faith is probably even more important than gold. Some religious buildings have great work of art slots, or even tourism, and you should at least strive to get those in all your cities ASAP.
2.
Many small cities with lots of religious buildings. For this victory type, food will actually not be that important. You'll want a few big cities, and sure, they'll need food to grow. But gold will actually be
their most important main resource, while either gold or production will be most important for your small cities, depending on if they have their religious buildings yet and which of the 2 resources are more readily available to them. No matter what, though, faith will be the single most important resource for every city the whole game. Food and even science both remain unimportant, though science will be needed for more advanced military to prevent attacks, while food is still somewhat important in your main 3 or 4 cities to keep them huge producers of gold and military units (when needed).
Domination victory is the most diverse victory of all, and can be done in about 4 different ways, going for victory in:
1.
The early game. This will be civilizations like Aztecs, Zulus, and Iroquois. Civilizations that are strong militarily early, but fall off in the mid-game. For this victory path, you can pretty much ignore food, focusing on production and gold to get your units out as quickly as possible. Culture and faith are basically useless, though they can give you slight benefits if focused in large chunks quickly very early. Science is only important before you get your needed tech for your units, if any at all. Food is moderately important -- as it always is -- but doesn't need to be focused on too heavily.
2.
The mid game. These are civilizations like the Mongols, Arabia, Spain, Greece, England. This is probably the rarest victory path of all 8 I'm going to talk about here (4 domination, 2 culture, 1 science, 1 diplomacy) but it's the one I'm 2nd-most-experienced with, and it's very very fun -- especially with the Mongols! (Note that some of these civs are versatile and can go for other victory types and only use their units for defense while they pursue other victories, but they can win domination victories as well, and that's what this paragraph addresses.) For this victory path, both food and science remain very important in the early game, until you have researched the necessary tech. Culture is moderately important to help give yourself an early boost. Faith can be ignored. Gold can be somewhat important if you plan to upgrade a lot of units as soon as you research the tech, but it's not necessary, and depends on the civ. Production is less important early on, but as soon as you research that tech it suddenly becomes the most important resource. Crank out those units! And once that tech is researched, food and science become much less important.
3.
The late game. These are civilizations like America, Germany, Japan, India (joke). This is the victory path I have the least experience with, as I don't find it that appealing and to be honest, it's probably not even that common. Anywho, these (bad) civs are focused on the late-game. They turtle the game and try not to lose while they wait until they can unleash their fury on their world. They are very similar to science victory, focusing a lot on science and food early, then production later to be able to build their UU's once they finally get them. Once the required techs have been learned in the late game, they can stop focusing on science. They can probably stop focusing on food even sooner, as it generally has diminishing returns anyway. Culture and gold remain moderately important, and faith isn't the least bit important.
4.
The whole game. This can be any civilization, really, but especially would include Babylon, Maya, Sweden, and the Ottomans. Why not just take your sweet time winning? That's what this victory path entails. Take a few cities here and there, and keep fighting all game. Civilizations good at this are ones that have either high science or UU's/UB's in multiple eras of the game. Basically, you want to focus science to stay ahead of your opponent and get to your desired techs ASAP. Production remains the most important resource all game, closely followed by food. Faith and culture are useless, and gold is somewhere in the middle.
Sorry for the very long reply! I didn't intend to write this much, but well, it happened. I hope it helps you a lot and gives you an idea of what you need for any situation! Let me know if you have any questions.