Once you have the airship, fly to Gaia, a town surrounded by mountains on the northmost part of the map.
From there fly west and south(a bit) until you find a castle. To land near the castle, you'll have to go north of it, over the water, and onto the regular colored land. Then you have to walk around it to the castle.
Beat the challenge there, making sure to open all the chests. The last chest you open on the second floor should have something called a Rat's Tail in it.
Once you have that, go back to your airship, and fly south and west until you come to a group of islands. Each island will have a hole on it. Two of the islands will have two holes.
Go into the RIGHT hole of one of them. If there is lava and an elf guy here, go back out. Don't continue, it will put you in a tedious, unescapable Soul of Chaos Dungeon. If you go into that one first, look for the other one. If you go into the right one, follow the halls onto each set of stairs.
Eventually you will come to a throne with a large dragon on it. This is Bahamut! If you have the rat's tail, talking to him will make him upgrade your party members' classes.
The upgrades:
Warrior-->Knight
Theif-->Ninja
Monk-->Master
Red Mage-->Red Wizard
White Mage-->White Wizard
Black Mage-->Black wizard
There are some spells unique to White and Black wizards. Ninjas can use levels 1-4 black magic. Knights can use levels 1-3 white magic.