Raids: Labyrinth of the Ancients (it's super easy), Binding/Second Coil of Bahamut (Second Coil is the hardest thing ingame atm)
Dungeons: there are tons, a few of them are hard-mode versions of story dungeons, which are still pretty neat.
Trials: Hard-mode primals (you do three of these anyway as part of your relic weapon quest), stuff like Chimera and Hydra (also part of the quest), Gilgamesh, Leviathan etc. And then there's extreme-mode versions. :D
Nearly all of them drop gear of all kinds - just pay attention to to the item level, item level goes all the way up to 115 at the moment. And those tomestones of mythology & soldiery can be traded in for equipment too.
You unlock pretty much everything through quests so they'll tell you some of this stuff anyway. First thing you want to start is relic weapon quests if you haven't yet. A lot of this stuff starts in Waking Sands if I recall.
Finished with Turn 2 of Coil forever. Finally got all my tank gear!