After completing the game, there are two new dungeons available immediately.
Dungeon
How to Unlock
Chocobo's Memories (Standard Dungeon)
Accessed through the Chocobo statue in the park
Croma's Future (Special Dungeon)
Accessed through Croma in Stella's House
Easy max equipment
After you help Freja get her memory back, go to her place, and have her hone your equipment to make it stronger. Before you start to hone, bring at least a few thousand gil, and save the game. When you get to the point where it says that it is at the maximum level and you can still hone it, you may want to save the game again; when Freja tries to hone it past it the maximum level it will have a greater chance to fail. If that happens, reset the game, load the saved game, and repeat until she makes a perfect hone. This may require a few attempts.
Easy Thief job
To easily get the Thief job, have at least the level 2 Black Mage. When you pick up the rare item from Merchant X (Thief's Memory) inside the Guardian of Flame dungeon (the glowing ring on floor 11 that will either take you to a duel or to a shop), walk away from Dungeon Hero X until he is close enough to use Sleep on. Then, run to the stairs. Try to run around the south side of the shop to make this easier. If Dungeon Hero X is gaining power, move two steps to the side of him. His beam is three spaces wide and goes very far. Keep Dungeon Hero X asleep all the time, if possible. This may take a few attempts, but it is much easier than any other way. If you get Dungeon Hero X asleep at the very start, you may be able to run directly to the stairs without going south around the shop, but this is much riskier. Since Dungeon Hero X can move much quicker than you, try to stay as far away from him as possible, but also making sure he is asleep.
Jobs
Successfully complete the indicated task to unlock the corresponding job:
Black Mage: Reach 10F in Guardian Of The Flame.
Dancer: Enter "Pirouette[Music Note]" as a case-sensitive Romantic Phrase.
Dark Knight: Defeat Croma Shade in 30F in Guardian Of The Light.
Dragoon: Successfully complete Meja's Memories.
Knight: Successfully complete Freja's Memories.
Ninja: Successfully complete Volg's Memories.
Scholar: Defeat the four elements in 20F in Guardian Of The Water.
Thief: Steal the item behind Merchant Hero X in one of his dungeon shops.
White Mage: Successfully complete Pastor Roche's Memories.
Jukebox in Dardola's Dineing
At some point in the game (I believe around Chapter Four) Head to Dardola's Dining and talk to Dardola (the guy with the green hair). The Bell will ring and he will lose his memories. Hop in his head and complete the dungeon. After this, He will make the jukebox in his shop available and you can now listen to the music in the game!
Additional Dungeons...