A few other things to consider too:
More luxury variety = more happiness.
You can achieve this by careful city placement and having workers prioritise working luxuries before anything else.
You can also make sure you expand towards luxuries faster by building / Tradition-acquiring monuments early, and by buying luxury tiles that look at risk of being swallowed up by other borders.
You can trade luxuries with other nations, though in the early game you probably want to be getting money rather than happiness.
Finally city states are probably the biggest source of unique luxuries, Quests are the most cost efficinet way to win them over.
Many buildings boost happiness.
A Circus is cheap and has no maintenance and gives +2 happiness. Some other resource-based buildings also give happiness (like Stone Works) but less so than Circus.
As noted, religious buildings often give happiness, and Pagodas are best for this. Small caveat: pagodas cost you faith you could be spending on missionaries, so are best if you intend your religion to be one that enhances your empire, rather than one that gains bonuses from converting the world.
Collosea give happiness too and have a maintenance cost, but once you have one in every city, you can build Circus Maximus, which will fix happiness for some time. Speaking of which:
Many Wonders give happiness.
Notre Dame is the big one, but there are others. Eiffel Tower is one I like a lot myself in the late game.
Circus Maximus is a National Wonder, so you don't have to race for it.
A lot of policies give happiness
Especially that Tradition one, which most people take as soon as they can. Once you develop your game you can play around with different openings, and there's big discussions about how best to work Liberty on this site, but the four city Tradition opening is popular because it is so strong and easy to play.
Ideologies give a TONNE of happiness
That is, if you're playing with BNW in place. Generally this is the stage of the game when mass happiness leads to massive growth and war opportunities, and a victory run becomes a lot easier. Speed players don't tend to get this far, but for you and me, Ideologies mark the turning point of the endgame starting.
The real trick here isn't just keeping happiness positive, its keeping happiness positive while juggling everything else the game wants you to do. If you're building a Coloseum you're not building a Knight. If you're teching happiness buildings, you're not teching science buildings. If you're working your economy, you're not working your military.
Generally a good goal is to stay positive in happiness almost all the time, and if you're unhappy to never get to -10 or worse. Also, you want to build a nice buffer of happiness that will let you go to war and capture or raze cities at some stage.
Happily, on Prince level you can expect to be better than the AI at working this balance, and can expect to be a lot better at warfare.
Re: avoid growth button, never ever use that. Its far better to micromanage your tiles and to set things up so growth comes at the right time. If nothing else, you can almost always switch a city's focus to make growth take so long that you'll have fixed happiness by that stage. If you do things this way, you get benefits in place of the food, whereas if you just avoid growth it'll just stop your food reserves at full, and not roll you over in population: something you're better off achieving manually.