Really the main thing to care about is getting allied with cultured city-states.
The culture they give you far outweighs what your cities can generate early-on.
So that means scouting the map as fast as you can and then checking to see what quests you can do. Once you get spies, steal 2-3 techs and then put them in the cultured city-states to attempt a coup or just to rig elections.
This by far is the best and cheapest way to increase your culture generation while at the same time denying it from the AI.
2nd is trying to win the world's fair. Again not only does it double your culture for 20 turns, but it also doesn't let the runaway AI double HIS.
3rd- Try to build the oracle. Most games if you build it before turn 90 you should get it. When you start it, watch the other capitals to see if any other AI has started building it.
4th- Build some archaeologists. If your culture sucks in the late-game you might be generating like 40-50 a turn. Antiquity sites can be worth 4-6 culture. Get 2-4 of them and you've just boosted your culture by 20-40% for much less than the build time of crappy culture buildings which I only ever build when going culture victory with the 50% discount from aestethics. Even then they're bad! +1 culture is so crap compared to what cultured city-states make, it's ridiculous.