The first dungeon is unlocked through the main scenario quests and you will find that this is the case for most dungeons in this game. The first dungeon is Sastasha which is a level 15 dungeon. As far as the chocobo goes you will have the option to get one upon reaching level 20, completing the first main scenario trial fight, and joining a Grand Company (and doing the quest that you get access to upon joining the Grand Company). My best advice is to just continue through the main scenario and you will eventually unlock these.