The thing to remember about this game is EVERYTHING is unlocked through a quest. If you don't have access to something either you need to move your main story quest forward or you need to move your class quest forward. But the quest level for certain things I can remember off the top of my head are as follows:

LvL 10: Able to pick up crafting and alternative classes (through class quest) - Chocobo Porter Unlock (through main story) - Duty Finder/Guildquests/Levequests (through their respective quests, e.g. in Horizon if questing near Ul'dah)
Lvl 15: Regular dungeons (required to be completed to move the story forward) - Class special ability unlock (class quest) - airship travel (main story)
Lvl 20: Mount unlock (through main story/quest), Pick your grand Company (through main story)
Lvl 25: Ability to create a Free Company
Lvl 30: Able to select a Job (Black Mage, Scholar, Warrior, etc) (Through class quest, has other requirements)

There are a bunch of little things that also unlock as you go through the earlier levels like Recommended List (shows nearby quests, hunting log etc), inns, your hunting log, minions. I don't remember the exact levels for these things but they unlock as you go.