Now how do you get into the level-50-only dungeons? I visited Amdapor and the Pharos in person, and there was no option to enter or to use Duty Finder to enter. Helpful fellow gaffers in chat said that there are people out in the world who will tell youabout these places as part of a quest; how are you supposed to find them? Eorzea is a prettty big place... or do I have to finish the main story before those quests will show up? Because I'd gladly do the rest of the story first if that's what's requierd.
Also, I finally found a way to dispose of the large number of unsellable Aetherial (pink icon background) weapons I'd been picking up in the dungeons: the Immortal Flames will give me seals for them. But is there a better-compensated way to unload them?
All level 50 dungeons minus Castrum Meridianum, Praetorium, and Sunken Temple of Qarn are optional. To gain access to these dungeons, you will need to do certain quests.
Pharos Sirius quest is given by NPC Diamanda in Aleport, with quest name "Sirius Business".
Amdapor Keep quest can be picked up from a fellow called Nedrick Ironheart standing outside the Waking Sands (where your story friends constantly are) in Vesper Bay, Western Thanalan.
Ideally, you can equip those pink background weapons/armor and spiritbond with them to 100%, at which point you can convert them into materia. Otherwise, the only way to get rid of them is at your Grand Company (you can offload them at the GC HQ, or at the various GC Leve people you see at certain camps around Eorzea). If you cannot equip the gear (it's for a class you don't have), then offload them. The seals can buy items, in your case being BLM or WHM you can spend them on Potash to bother some GAF FC crafters into making you one of these:
Which is decent equipment to go into the hard mode primals with. The staff is white mage only though.