After obtaining a ship in Bloomingdale, you can sail to Gleeba and meet with the Queen.
After passing through the Djust Desert to reach the central city of Gleeba, you can get some new recipes and a new party trick. Then you can help Queen Voluptua.
Get New Recipes
As soon as you enter the city of Gleeba, go into the house to the left of the entrance. Check on the bookshelves there to learn some new alchemy recipes.
Enter the Mirage Mahal
The palace in the north part of Gleeba is the Mirage Mahal. It's only open to the public during the day, so be sure to visit it then.
On the first floor of the Mirage Mahal, go to the northeast and northwest rooms and check all of the bookshelves. You will learn new recipes and a new party trick.
Help the Chancellor
Go up to the second floor of the Mirage Mahal and go to the area in the top middle where a man is walking near lion statues and a throne. Talk to him and agree to help. He will ask you to find the Queen's missing lizard.
Find the Lizard
If you talk to people in town, you will get some hints about finding the lizard. To find it, go to the area inside the city that is just west of the Mirage Mahal. There is a tree in the shade here. Use the Clap party trick to scare the lizard out of the tree, then chase after it and press A when standing near it to catch it.
Bring the Lizard to the Chancellor
Go back to the second floor of the Mirage Mahal and talk to the Chancellor to show him the lizard. After a meeting with Queen Voluptua, you have to find a way into the bath to retrieve the Fygg.
Find a Way Into the Bathing Chamber
Go down to the first floor and talk to the person guarding the door to Queen Voluptua's private bathing chamber. Then go down through the archway across from the bathing chamber door. Talk to the maiden there to find out about someone who might know a secret way in.
Go up to the top floor. You will see a treasure chest on the other side of a fence. Exit this room through the door to the south, then go around the back of the dome and walk into the north wall to get into the secret chamber. Open the treasure chest to get the Magic Key. This will allow you to open the locked purple doors that you might have seen around the world.
Now go back outside and talk to the man sitting on the fountain. Tell him that you want to get into the private bathing chamber. After that, check on the middle of the fountain and say Yes when asked if you want to jump in. Watch the scene, and then jump into the well in the bathing chamber to enter the Plumbed Depths.