(Oops posted late lol)

You'll gain access to the other areas through your Main Story Quests generally around level 15. Since your a Thaumaturge, for a cure spell, I would suggest Arcanist's Physick ability (also a healing spell) instead of Cure since it will be able to be to keep it on your hot bar when you unlock Black Mage at level 30 while you'll lose Cure.

You will be able to start using mounts once you join a Grand Company, which you'll be prompted to do around level 20 I believe in your Main Story quests. When you join the Grand Company, there'll be a blue quest at the Grand Company called "My Little Chocobo" that will allow you to use mounts.