The Chocobo is part of your "Play Plan". When you open the main menu, you should see the little Chocobo icon on the lower right of the screen. It lists the bonuses and such that the Chocobo will reach in the future. Basically, if the Chocobo catches a carrot, he will give you an Exp boost for the next battle. If he reaches a treasure chest, you will get a new accessory that has a chance of breaking, but boosts your experience. And, depending on how many battles you have fought that day, he has an increasing chance to hit a lucky note, which will confer a lesser Exp bonus than the carrots, but you can get one of those every battle if you build the lucky chance up to 100% (after 100 battles that day).

Cecil transforms when he performs his HP attacks, or freely while in EX mode. If he performs an HP attack in the air, he shifts to his Paladin form, if he performs an HP attack on the ground, he becomes a Dark Knight.