For now, play it safe and follow the general advice posted already in this thread. Later on, when you have a better understanding of how the game mechanics work and have a lot more wins under your belt, you can get more creative in how you deal with these kind of situations. There is no single 'best' way to go about it, as every CiV game is different, and every gamer is different.
I've done all of the different variations already mentioned, in different games. But now I approach every war and every 'surrender' scenario with a sharper eye to benefiting my current civilization's victory condition. Whether you annex or puppet, may depend a lot on whether you are going for domination, science, culture, or whathaveyou. Same with whether you raze most or all of the cities you end up getting. A lot depends on the cities themselves- their locations, resources, luxuries, population, whether I need any of them for bases for expansion, or forts for defense, or if I'd be better off with a bunch of razed areas to buffer me from my enemies for a while.
And of course, the happiness thing... in G&K, it shouldn't be terribly hard to keep at least a moderate amount of banked happiness for your CiV... if you are in low single figures going into a war scenario where you expect to end up with captured cities, that is way not good. Especially 'gifted' cities when your enemy caves in... they have much higher populations than invaded/captured cities, and cost you a lot more happy's. At any rate, once I win a war that way and end up having to decide what to do with multiple new cities, I'll usually accept them all, and puppet them all right off. The next turn, you can decide which ones you want to keep, annex later, sell off, raze, leave puppeted, or whatever.
Any major city with good resources, decent population, and maybe even a Wonder or two, is a good candidate for eventual annexation. Especially when they have a lot of mines and can be turned into a high production city. Cities with mostly grass and plains and such, I leave as puppets and send in the peons to turn them into gold factories dotted with trading posts. Small, ineffectual cities that the AI spammed in stupid places, with no important resources and limited future benefit from letting them grow, I'll raze. And if I'm going culture, I'll probably just leave them all puppeted, so as not to screw up my culture numbers.
And, in some circumstances, you may just want to not accept all of the cities offered... you can refuse all or some of the cities that are offered to you, in the armistice screen. For instance, if the civ you just beat is not a huge threat to you, and you want to leave them with enough resources to defend themselves (and you) from another bigger threat nearby. If you are not going for domination vic, this can sometimes be a valid strat- you know the putz you just beat will likely never be a serious threat to you, but you want them to at least remain a half-decent speed bump for that runaway civ on his other side.
There are a bazillion factors that can influence how you deal with captured cities, and you'll get the hang of it, with experience