With exception to those that are unlocked by story progression

Halatali (20) - Quest from Nedrick at Vesper Bay
The Sunken Temple of Qarn (35) - Nedrick
Cutter's Cry (38) - Quest from Sibold in Ul'dah - Steps of Thal
Dzemael Darkhold (44) - Quest from (someone) at Observatorium in Coerthas
The Aurum Vale (47) - Nedrick

50 dungeons:
The Wanderer's Palace - Allene, Vesper Bay
Amdapor Keep - Nedrick. Must have completed Aurum Vale
Pharos Sirius - Aleport, Western La Noscea
The Lost City of Amdapor - Conjurer Guild, Gridania
Hard mode dungeons (Copperbell, Haukke, Halatali, Brayflox) - Mor Dhona, Seventh Heaven bar, various NPCs.

Trials:
Battle on the Big Bridge - Unlocked through the Hildibrand quest line, starting in Ul'dah at 50 (The Rise and Fall of Gentlemen)
Hard mode Primals - Waking Sands, Urianger. Ifrit > Garuda > Titan
Extreme mode Primals - Waking Sands, Urianger. Garuda > Titan > Ifrit > Moogle/Leviathan
The Minstel's Ballad: Ultima's Bane: Seventh Heaven bar, Mor Dhona

Raids:
Labyrinth of the Ancients (Crystal Tower part 1): series of quests starting in Mor Dhona
Binding Coil of Bahamut: Urianger, Waking Sands
Second Coil of Bahamut: Urianger, Waking Sands. After beating Turn 5 of Binding Coil.