You can only serve 1 GC at a time. You can change your allegiance to one of the other two but you lose all progress and ranks; and if you decide to rejoin your old GC you start from the bottom. The only thing you can do "quest-wise" for GC is the GC leves up until level 40 when the actual quest lines open up. As far as the leves are concerned, you can do GC leves for any of the three GCs but you get extra credit for doing leves specific to your GC. Personally, I find the leves to be lacking. You get way more seals from vendoring gear and running FATEs for the effort involved. I save my leve allowances for crafting leves.
EDIT: The first and foremost task you should accomplish after joining a GC is to acquire 2,000 grand seals and then return to your GC and get your Chocobo! Life is so much better with a mount!
Edited, Sep 10th 2013 11:40am by Azoria
"On a mountain of skulls in a castle of pain, I sat on a throne of blood. What was will be, what is will be no more. Now is the season of evil!"
~Vigo the Carpathian