Imagine your empire as one large megacity, and each city is a building. Just like the buildings of your city support each other, the cities of your empire support each other.
Your border cities need culture, your interior cities do not.
Your largest cities need health/ happiness, your smallest cities do not.
You have to ask yourself, what does each city bring to your empire? Even with libraries, etc... the majority of your beakers/ coins/ culture is going to come from your key cities, not your fringe cities. Once you've built the necessities, it's more important for your fringe cities to be building Wealth (so you can run at 100% beakers or culture), Units to raise your power and defend, or Culture to expand your territory/ fight off expansion.
Basically, what each city is adding to your empire is:
An extra temple for Cathedrals in your key cities.
An extra courthouse for Forbidden Palace.
An extra university for Oxford.
An extra bank for Wall Street.
Ok so here is some breakdown:
Each city should have (in priority):
Granary for whipping
Forge to increase production
Levee to increase production, if applicable
Temples to build cathedral and to get bonuses from Sistine Chapel, University of Sankore, Spiral Miranet
Courthouse for Forbidden Palace and maintenance reduction
Library for University and culture
University, for Oxford University and culture, although after you get Oxford you don't build this except for high commerce cities
Banks are of low priority because Wall Street is so late.
In your border support cities, you will want:
Theater for culture and Artist specialists
In your interior support cities, you will want:
Market and/ or preferably Grocer, to run Merchant specialists
Jail, Intelligence Agency and Security Bureau when you have cash overflow/ production to spare
In your sea cities you will want:
Customs House to increase trade yield
Harbor to increase trade yield
Lighthouse to increase food production, but only if it's more advantageous to work coast tiles instead of Merchant/ Artist specialists
In your capital you might build:
Wall/ Castle, but only if you get Engineering very early, and only if you have stone
Everything
In cities along a hostile border:
Barracks
Stables
Units with some production queued up to be whipped, and perhaps some happiness buildings as well
Observatories: Only build in your capital and high commerce cities
Colosseums: Don't build them unless it's your UB or you whip too much
The key concept is raw commerce. You want your raw commerce to be as high as possible, since it can be directed into Beaker or Culture, and Coins when you run low on cash. The multipliers such as Market/ Library are useful, but only in larger cities. Thus buildings that increase raw commerce such as Harbor/ Custom House >>> Market/ Library. By building wealth/ running Merchant specialists in your small cities instead of building Universities, you will be able to run at 100% beakers more often and therefore be much further out ahead.
Production is the most critical limiting factor to your growth. You need to prioritize your limited whipping ability by spending hammers on what is most important.
Cheers,
Dai