Well, technically the long answer would be that you have to play through the story quest line. At around level 20 in the story mode, you're given the option of which GC to join.

As an added after thought to what Austin said, joining a GC doesn't happen automatically at LVL 20. I know a few people that FATEd their way to lvl 20 and were asking, "Where's my mount?" Also, getting your chocobo as a companion to fight alongside you in battle doesn't happen until about lvl 25-30 in the story mode.