Do the Main Scenario Quests, it will unlock access to other cities prior to the first dungeon. After joining a Grand Company via a the MSQ, you will be able to purchase a chocobo license with Grand Company seals from them. You can obtain those by doing FATES once you are a member of a Grand Company or by doing the GC hunt log you will receive on joining. Around level 30 there is a quest at Camp Tranquil in the Black Shroud where you can learn to summon your chocobo to fight with you. Most things in the game are unlocked or triggered via the MSQ or class/job quests, so it is a good idea to keep up to date with them.