There's a fairly straight forward progression of dungeons from the start of Final Fantasy 14: A Realm Reborn to level 50. However, once you hit level 50, there are a ton of dungeons and trials you can play through, but aside from a few trials, there isn't much clarity as to what you're supposed to run first, or if there's any order at all. The kind of gear you're looking for plays a big role, but to make an informed choice, you need to know what each dungeon and trial offers.

Hard Mode Primals

The Hard Mode Primal battles are primarily meant to be the path toward a relic weapon. Ifrit drops level 60 weapons, Garuda drops level 70 weapons and Titan drops level 80 weapons. You need to defeat all three (Ifrit, then Garuda, then Titan) before you can complete the relic weapon quest to obtain a level 80 weapon that can be upgraded to level 90. It's also required to defeat all three Hard Mode Primals before you can run any of the Extreme Mode variants. Hard Mode Leviathan is an extension of the story quests and therefore not required to gain access to the any Extreme Mode battle except Leviathan.

Extreme Mode Primals

Once you defeat all three Hard Mode Primals, you gain access to the Extreme Mode variant of Garuda. The three main Extreme Mode Primals offer high level accessories. Garuda drops level 90 rings, Titan drops level 90 earrings and Ifrit drops level 90 wrist pieces. Defeat all three in that order, and Hard Mode Leviathan (as well as the story quests that go along with it) to gain access to Extreme Mode Leviathan. The Extreme Mode variant of Leviathan drops level 95 weapons and shields, which can be upgraded to level 100.

Thornmarch (Hard and Extreme)

The Hard Mode variation of Thornmarch is a pit stop as you're attempting to achieve a relic weapon. It drops level 75 weapons for all classes, which means the weapons are better than what you receive from the Hard Mode versions of Ifrit and Garuda, but not as good as the weapon drops from Hard Mode Titan. However, Thornmarch is generally considered much easier than Hard Mode Titan, so if you're having difficulty with Titan, complete Thornmarch to get the level 75 weapon, then try Titan again. The Extreme Mode Thornmarch drops level 100 neck pieces to go along with your level 100 weapons from Extreme Mode Leviathan.

Relic Weapon Progression

Relic weapons start at level 80, and can be upgraded to level 90. Once your relic weapon reaches level 90, you can then begin the process of upgrading it even further. This should be one of your primary focuses once you hit level 50, because it requires a bit of time to complete the initial quest. You must defeat the Chimera, Hydra, the Hard Mode versions of Ifrit, Garuda and Titan, then collect 900 Mythology Tomes to upgrade it.

Ultima's Bane

If you're having trouble with the Extreme Mode Primals, or you wish to spend your Myth Tomes on main armor instead of accessories, the Ultima's Bane battle rewards you with level 80 rings and necklaces. The battle is more difficult than the Hard Mode Primals, but easier than the Extreme Mode variants. It’s also easier, since the most recent patch introduces Leviathan.

Dungeons

Running through the various level 50 dungeons will net you level 70 Darklight gear, and award you with Myth Tomes to purchase level 90 gear and Soldiery Tomes to purchase level 100 gear. You can also run through the Crystal Tower (Labyrinth of the Ancients) to obtain level 80 gear. The first Binding Coil of Bahamut drops level 90 gear and a level 95 weapon, while the second Coil drops level 110 gear and a level 115 weapon.

All of the endgame dungeons aside from the first and second Coil are designed to help you gear up for Coil runs. Once you complete the first Coil, you should be geared enough to run through the Extreme Mode Primals and start the second Coil. The second Coil offers the highest level gear in the game and is currently the most challenging dungeon available.