You don't receive the warmest welcome in the Underworld. The only place to go is this, the Dwarven Castle. Most of the castle is off-limits when you first arrive, with the exception of the inn and Item Shop. Speaking to the king will begin a cutscene where Cid will leave the party, who then fight two bosses. Rydia will join the party during the latter fight, and you don't have time to re-order it after Cid leaves, so put him in the back row (and unequip him if you want).
The first boss is Calbrina, who will begin the fight as three Cals and three Brinas. The Brinas are much weaker, but regardless, after killing a number of them they will combine into Calbrina, a single large Monster. Kill it quickly, because if you take too long it will split back into its six smaller Monsters, even if you already killed most of them. Berserk will help prevent this.
The second fight is mostly a Plot Battle, but you can certainly lose. Golbez will appear and kill most of the party. If you're lucky, you can get Kain to Jump before the fun begins, and he will be spared. Otherwise you'll be limited to Cecil and newcomer Rydia. Don't try to heal everyone, just attack with Cecil and have Rydia summon Titan to end the fight quickly.
After all the fighting, explore the castle and collect all the Treasures. You'll likely want to switch Cecil's Flame Sword for the Ogre Axe you find, since many enemies in the Underworld resist Fire. The Weapon and Armor Shops also contain a lot of nice upgrades. Rune Rings are particularly nice, arguably the best easy-access arm gear in the game for your mages. If you walk between the counters you'll find the Rally-ho Pub, and from there you can walk through the southeast wall to find the Developers' Room. There's nothing vital in here, but it is pretty fun.
When you're ready to leave, head for the secret passage in the basement. There's a pot that restores HP and MP near the exit, so you can use this instead of the inn to save time and money while exploring the Underworld.
After returning from the overworld, head back here and speak to King Giott to have your airship upgraded so it can fly over magma. This will open up the rest of the Underworld for exploration.
You'll come here one last time, for another airship upgrade, after finishing the Sealed Cave. This will allow you to permanently open the route between the overworld and Underworld, and trigger the appearance of the Lunar Whale when you travel to Mysidia, not to mention opening a number of side quests.
The first boss is Calbrina, who will begin the fight as three Cals and three Brinas. The Brinas are much weaker, but regardless, after killing a number of them they will combine into Calbrina, a single large Monster. Kill it quickly, because if you take too long it will split back into its six smaller Monsters, even if you already killed most of them. Berserk will help prevent this.
The second fight is mostly a Plot Battle, but you can certainly lose. Golbez will appear and kill most of the party. If you're lucky, you can get Kain to Jump before the fun begins, and he will be spared. Otherwise you'll be limited to Cecil and newcomer Rydia. Don't try to heal everyone, just attack with Cecil and have Rydia summon Titan to end the fight quickly.
After all the fighting, explore the castle and collect all the Treasures. You'll likely want to switch Cecil's Flame Sword for the Ogre Axe you find, since many enemies in the Underworld resist Fire. The Weapon and Armor Shops also contain a lot of nice upgrades. Rune Rings are particularly nice, arguably the best easy-access arm gear in the game for your mages. If you walk between the counters you'll find the Rally-ho Pub, and from there you can walk through the southeast wall to find the Developers' Room. There's nothing vital in here, but it is pretty fun.
When you're ready to leave, head for the secret passage in the basement. There's a pot that restores HP and MP near the exit, so you can use this instead of the inn to save time and money while exploring the Underworld.
After returning from the overworld, head back here and speak to King Giott to have your airship upgraded so it can fly over magma. This will open up the rest of the Underworld for exploration.
You'll come here one last time, for another airship upgrade, after finishing the Sealed Cave. This will allow you to permanently open the route between the overworld and Underworld, and trigger the appearance of the Lunar Whale when you travel to Mysidia, not to mention opening a number of side quests.