Obviously, for a start - build colosseums/theatres/stadiums. Have as many different luxuries you can have. Tip: build city on top of luxury to have access to it without the use of worker; also, city square cannot be pillaged.
Early game: I usually go with Liberty SP - basically everything in it asap. Connect cities to the capital. Liberty finished = Great Person of choice, which essentially means a world wonder of choice. Choose great engineer and hurry up/build Notre Dame. +10 hap. early game really helps. Chichen Itza's +4 is not bad either. If you have Theology, hurry Hagia Sophia and get another GE from it - then build ND.
Forbidden palace is decent, but somehow I usually fail to build it (and get it later on during conquer phase
I usually go with Piety/Organized Religion - you have to build as much Monuments/Temples/Monasteries as you can.
Late game, Order helps - +1 per city is good. Eiffel Tower is great.
What can really make a difference is Honor/Professional Army SP. Walls/Castle/Arsenal/Mil. base don't cost upkeep, so you can have up to +4 happiness per city - with no additional cost. So build (buy) Walls/Castle etc. everywhere, not just in border cities.
Mind also the "Military Caste" - have unit in each city for +1hap/+2culture. I usually use this to get out from "very unhappy" situation.