There's waaaaay more to it than this. I'm really tired right now so I might miss some, and I still haven't figured out all the nuances of the game but here's a quick list.
Health:
Terrain: Certain terrain has negative and positive effects. Fresh water and forests have positive effect and jungles and floodplains are negative. Building your city in the middle of a huge floodplain will always cause you problems. Chop down jungles as soon as you can and don't chop down all your forests.
Terrain Bonuses: As far as I can tell, having a terrain bonus that improves health, like corn or wheat or whatever, connected to your trade network will improve the heath of all your cities. Having an improvement like a grocery built will increase this benefit. It's like the health version of the marketplace in CivIII.
City Improvements: From what I can remember aqueducts, groceries, harbors and hospitals all improve city health (I know there's more) and so do a few wonders. Production improvements like forges, drydocks and factories all have negative health.
Happiness:
Terrain Bonuses: Luxury resources are the best way to go. Having these tiles improved and roaded is a must. Unfortunately, you have to wait to research specific techs for some of them before you can build the neccessary improvement. I think you need Calendar for plantations and Monarchy for Wineries, which both come a little later in the game (especially if you're playing at epic speed) around the time where they're REALLY needed.
City Improvements: Theaters, temples etc. and a few great wonders can take care of unhappiness problems fast. I can't remember at the moment, but the marketplace doesn't increase happiness from luxuries like it did in CivIII. Is there any other improvement that does this?
Religion and Civics: The big difference in this game is the religion benefit, which can really affect happiness in this game in a huge variety of ways. For one, you can only build a temple in a city with an established religion. The Civics make a big deal in this too. Combining your religion and civics just right can take care of your unhappiness problems completely.