The Dungeon of Trials is just east of Temple Town.
It's a pretty cool little dungeon. Each time you enter, every floor of the dungeon is randomly generated, and each succeeding floor is guarded by one of the game's bosses. (For example, the stairs to the second floor are guarded by the boss from the E. Tower, the first boss in the game.)
Because it's located right next to Temple Town, the Dungeon of Trials is an excellent location for grinding through all the different job classes as they become available.
It's also the only place in the game where you can obtain the Odd Objects that the Alchemist in Temple Town uses to combine into different items, so if you want to experiment with that, you'll want to spend a good deal of time in this dungeon.
But, aside from grinding and collecting Odd Objects, the dungeon has no impact on the story, so if you want to skip it altogether, you can. I've only included it here in passing because of its location next to Temple Town.