To get going with City States you need cash, that buys "influence". The amounts concerned are 250, 500 and 1000 gold. If you donate enough cash to meet the level required to be "Friends" you get a limited amount of their dedicated resource specialty (be it food, military, or [bizarly] "culture"). If you pass the top level of donation of cash then all the natural resources they have developed is added to the list and an enhanced level of their specialty resource. The influence you buy degrades slowly at around 0.75 to 1.5 per turn, depending on many factors. So you need to keep them "topped up" with cash every now and then - look on that aspect as a Bar Owner paying for "protection" to the local Mob (kind of thing), you want their help, you have to continue to pay, its not a one off donation.
Click on any City State name/icon etc, and read the screen that opens up, all will become clear. As stated earlier in the thread only one Civilisation can be Allies with a City state at anyone time. Pick your CS wisely in the early game (there are no restrictions on how many you can ally with if you have the cash - although at a point in time other civs start getting hissed off if you grab too many, ignore the hissing if your military is ok, but just be aware of the latter). If you grab one (bribe it) and its actually on another Civ's land mass, that Civ gets hissed off big time depending on whether or not you have a decent military.
Lots more subtle stuff to them, but whatever you do, in the game as it stands at present, dont ignore them, learn how they work, its worth the effort. In particular watch the Policy Tree "Patronage", that line of policies is essential to get the best out of City States.
In the game as designed at present they are an essential source of Food, Culture and Military units without which your chances of "winning" are reduced or considerably delayed (as such), many think they are in fact over powered.
Regards
Zy