WIP. This is part of a series of posts intended to serve as a quick reference for people looking for that first breadcrumb. If I miss anything, post away and I'll go ahead and edit this.
A lot of things will unlock with story progression. Ignoring these since they're quite hard to miss.
Level 50 required:
Ifrit / Garuda / Titan HARD: All three are a sidequest chain starting in the Waking Sands. Talk to Thancred / Papalymo / Y'shtola respectively
Gilgamesh (Big Bridge): Part of the Hildibrand questline. Start with Wymond in Ul'dah - Steps of Nald (x9,y8)
Chimera / Hydra: Relic Weapons quest. Nedrick Ironheart in Western Thanalan (x12,y14)
Ultima Weapon: The Ultimate Ballard - Alys in Mor Dhona (x21,y8)
Garuda /Titan / Ifrit EXTREME: Continuation of the Primal HARDs. Urianger at the Waking Sands. Repeatedly.
Good King Moggle Mog XII EXTREME: More Urianger Madness. Waking Sands.
Leviathan EXTREME: Urianger Returns. The darker, grittier sequel.
Ultros & Typhon: Further down the Hildibrand quests.
Ramuh EXTREME: NOT Urianger. I know. You should probably have a seat somewhere. Scarlett - New Gridania (x9,y11)
Gilgamesh (Big Keep): Hildibrand. Why did you not just do them all by now?
Shiva EXTREME: Urianger. The world is right once again.
Odin: Scarlett - New Gridania (x9,y11)