To begin the quest to obtain your personal chocobo, speak to the NPC above which belongs to the Grand Company you joined. You will be tasked with acquiring a Chocobo Issuance, which is purchased from your Grand Companies quartermaster, under the Materiel tab, for 2,000 Company Seals. These can be earned in a number of ways, including participating in FATE events, completing portions of your Grand Company Hunting Log, turning in supply (crafted) and provisioning items (gathered) to a Personnel Officer, or completing Grand Company Levequests at the following locations:
I personally found that just running around participating in FATEs provided the best source of Company Seals (~120 per FATE). If no FATEs are popping up, the Grand Company Levequests are solo and return about the same number of seals. If neither of those are your style, feel free to use whichever method you prefer.
Once you've accumulated 2,000 Company Seals, purchase the Chocobo Issuance from your Grand Company quartermaster and turn the quest in to claim your Chocobo Whistle!
Once you reach level 30, you will be able to teach your chocobo to fight along side you, either as a tank, healer, or damage dealer. You will find this quest "My Feisty Little Chocobo" hidden away at Camp Tranquil in South Shroud, from the NPC Docette.
Once you have it unlocked, you will need to grab some Gysahl Greens to summon it to battle. These are sold by the following NPCs: