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.
I'll cluster things out of difficulty order where things become available at the same time and in the same place/area. Turns out not a huge number of places to go anyway. I'll just make another post with a simple "go here, grab everything" approach.
Halatali: From level 20, "Hallo Halatali" Nedrick Ironheart in Vesper Bay
The Sunken Temple of Qarn: From level 35, "Braving New Depths" Nedrick Ironheart in Vesper Bay
Cutter's Cry: From level 38, "Dishonour Before Death" Sibold in Ul'dah - Steps of Thal
Dzemael Darkhold: From level 44, "Fort of Fear" Carrilaut in Coerthas Central Highlands (x24,y28)
OR "Shadows Uncast" from your Grand Company commander if you've reached Sergeant First Class
Aurum Vale: From level 47, "Going for Gold" Nedrick Ironheart in Vesper Bay
Level 50 required, almost all of these require you to have completed "The Ultimate Weapon".
The Wanderer's Palace: "Trauma Queen" - Allene in Vesper Bay. The only case where "The Ultimate Weapon" is not a requirement.
Amdapor Keep: "Ghosts of Amdapor" - Nedrick Ironheart in Vesper Bay
Pharos Sirius: "Sirius Business" - Diamanda in Aleport. You'll probably want to do "Why So Sirius" when you finish "Siruis Business" since its a pre-req. for Hard much further down the line.
Copperbell Mines HARD: "Out of Sight, Out of Mine" - Hugubert in Mor Dhona
Haukke Manor HARD: "Manic Manor" - Lauriane in Mor Dhona
The Lost City of Amdapor: "One Night in Amdapor" - E-Sumi-Yan in Old Gridania (Conjurers Guild [confirmation needed])
Halatali HARD: "This Time's for Fun" - Hugubert in Mor Dhona
Brayflox's Longstop HARD: "Curds and Slay" - Bloezoeng in Mor Dhona
Hullbreaker Isle: "King of the Hull" - Bloezoeng in Mor Dhona
The Stone Vigil HARD: "Blood for Stone" - Faillicie in Mor Dhona
Tam-Tara Deepcroft: "Corpse Groom" - Paiyo Reiyo in Ul'dah - Steps of Nald
Sastasha HARD: "It's Definitely Pirates" - Bloezoeng in Mor Dhona
The Sunken Temple of Qarn: "The Wrath of Qarn" - Hugubert in Mor Dhona
Wanderer's Palace HARD: "Not Easy Being Green" - Bloezoeng in Mor Dhona
Amdapor Keep HARD: "For Keep's Sake" - Lauriane in Mor Dhona
Thats all the pre-expansion stuff. Next time, Heavensward!