Build your cities within range of horses or elephants. That way, you can build circus for happiness. So Animal Husbandry is important. Try and get at least one wonder for happiness. I like the Eiffel Tower myself though that's later on. But there are others just as good. Be conscious of what policies you take that help happiness. If you go for science, then the two best policies for what you're looking for are civil society and democracy if you use specialists. They only come at Renaissance though.
At the start, for growth, the two techs that will affect growth a lot are Civil Service (extra

) along rivers and later, Fertilizer for extra

everywhere else.
Then you should probably build happiness buildings like the Colosseum.
BTW, one thing I used to do VERY wrongly is to just go for high growth/population for no reason. There is no point in having an extra citizen if you don't get anything from it. Most of the games I had a hard time with, this was the main cause of my troubles. I now only get a new citizen when it gave me an advantage for that happiness point.
I'll give you an example of when getting a new citizen is bad. If you get a new citizen and you're with one happiness less, but your production and food and pretty much everything else is the same, then you're in trouble. Sorry, the extra science isn't gonna be of much use if your city is stonewalling your entire civilization.
Until Civil Service, I'm looking for places to settle and getting monuments and building up an army.
Oh... forgot probably the most important point. Are you playing with any AI's? If so, and even if not, you gotta trade luxuries. You can't do it without luxuries. Sell all extra luxuries to the AI and trade whatever luxuries one for one you can, even buying them if necessary. Only if you need more happiness, but sell everything you don't need @ 240 or whatever you can get. At the beginning, sell open borders to AI's for 50 gold.
edit:
Saw you mentioned 7-12 range for high pop. This is later. So I'll assume you have Civil Service and Fertilizer. Civil society and Democracy policy are your friend. But for city growth, Aqueduct and Hospital are what you want. A Hospital gives 5 extra food, but no longer retains food. I think the Hospital is far better as it frees up 2.5 citizens, or 5 specialist if you have Civil Society. I've had cities with 30+ pop doing this, though I do get the buildings and improvements for specific tiles like grain tiles. Those will provide insane amounts of food. You also need to be able to select what tiles are worked by what citizens.