Happiness is usually only a problem in early game.
Basically what you want to do is sell surplus resources your not using to keep yourself solvent. (Have a 4 Horse resource but no plans on building any horse units over the next 30 turns? Then sell it for 180 gold. Or perhaps you want to build 2 horse type units but not all four then sell 2 of the horses for 90 gold.) And if you have a luxury but can't find an AI that has a different luxury but does have 240 gold, you sell that spare luxury. (More advanced is to look to sell for cash first and only be looking for a trade if you need additional happiness)
Make building Colosseums and all local happiness buildings that depend upon terrain such as a Circus a priority. Along with Circus Maximumus. (Additional 5 happiness)
Found a religion. If planning on being peaceful, then choose a follower belief with happiness bonuses; there are several to choose from and all you'll need is one.
If planning on Domination choose the religious founder belief and in addition pick a follower belief that's a building you can build in every city. (The same applies playing Austria)
Next, following completion of your first tree; choose Patronage followed by Ascetics as filler policy. Combine this with pledge to protect abuse to get free friendship with every city state on the map. This will get you +3 happiness for every merchant city state on the map; along with a massive food bonus to the capital and other benefits.
Playing peaceful this will allow you to grow your cities as much as you want while not having to build more than one Theater; and sometimes not even that.
If you are planning on conquering cities; raze the AI's junk cities and only keep the good ones. (And in this case you probably will need to build multiple Theaters.) For the ones you keep at all, initially puppet. If it's really good such as a capital, you can always annex after it comes out of resistance.
