You don't take sides as such, not by pressing a button at least. There are three main factions; Alliance, Federation and Empire. Then lots of smaller ones, some allied to the big three, some independant, some in the middle of civil war (usually Anarchy systems). You can check all these out using the galaxy map and selecting which faction or even which government type you want to search for.
Then, when you find something that suits you, start selecting missions from the bulletin board from that faction. Once you've completed enough, you will notice a friendlier welcome on your return, the station turns green in your targeting system and you will see in your right hand panel that you have gone from neutral to friend. Do more and you become an ally, after that I don't know, that's as far as I've got. Just be careful when out on your travels... if for example you are allied with an Alliance station and you're flying around in a Federation system and see a wanted pilot, don't just shoot without checking who he is. He may be wanted in the Federation but he could be friendly/allied with Alliance and not wanted there... you shoot him and you can lose your good rep with Alliance.