Upon reaching level 5, you can finally head to your first gym. It does not matter what color the first gym you go to is on your map--you will still be able to choose your team!
The different team leaders claim to have different methods of researching Pokemon. It does not seem that there is any technical advantage to choosing a team--but some speculation states that the Legendary Birds that represent each team’s color may come into play sometime in the future. This means that the color team you choose might affect the Legendary Bird you might potentially interact with (but again, that is complete speculation at this point).
Once you choose your gym color, you have the ability to train at that gym. If there is space available at that gym, you can leave your Pokemon there to defend your team. You'll also earn some bonuses.
You can also challenge the gyms of other teams. By doing so you can eventually have your team take over. You get to choose 6 Pokemon and with those 6 you get to battle the opposing team's hierarchy of winning Pokemon. If you challenge your own gym, you only get one Pokemon to fight your way to the top.
Make sure you coordinate with your friends on choosing a team, because once you pick your color, you cannot change your mind.
If the theory about Legendary Birds is correct, you would choose yellow Team Instinct for Zapdos, blue Team Mystic for Articuno, and red Team Valor for Moltres.
Aside from arguing with other people over who has the better team and potentially changing your entire wardrobe to display your team color, there are no observed differences between the teams. They are simply there to add an element of competition to the game.