Working through my first Civ V game now, though I actually bought it over a year ago I think. If I didn't have the difficulty set on an easy level, it would not be going well for me. Started the game surrounded by city-states with no access to horses or iron. So much for my old strategy of rapid early expansion.
I don't think I like having city-states in the game. Other than being a PITA, I'm not sure what purpose they serve.
Like shortylickens said, you can get bonuses from them if you please them (they randomly give out missions you can complete to gain favor with them) or provide them with gifts of gold/units (don't bother with units unless you are already planning to get rid of the unit, the favor you gain with them is far too small compared to the cost of the unit).
Being surrounded by city-states isn't the end of the world anyways. They provide a very nice buffer against AI civs who declare war on you (if you can get the city-state to 'ally' status, they will declare war on any of your enemies). Usually its a losing battle for the city-state, but you can either use that time to build up your own forces, or go fight with the city-state, using your combined forces to overcome a more powerful enemy.
Also, you can simply walk through city-state territory. It makes them angry if you aren't allied with them, but the anger wears off pretty quickly (it decays on its own over time), and you aren't likely to be able to afford to make them an ally in the early game anyways. City-states will never declare war on their own, so feel free to piss them off as much as you want if you don't foresee trying to make them an ally later in the game.
The bonuses, while usually quite small, can be very significant. Because of how modifiers work in Civ, you usually want to focus on a couple things and the means you fall behind in other areas. Every city-state bonus provides a very valuable type of resource. For instance, if you focus heavily on military, you may not have much time to build up your culture/faith infrastructure, but even a small amount of culture/faith can go a long way. Having a couple allied city states providing you with culture/faith helps make up that difference.
Theres a lot more to city-states, but I'd advise you not to turn them off until you get a better understanding of the game.