Final Fantasy IV/To The Moon

Fly to Mysidia
Once in the overworld fly immediately to your right and you will reach Mysidia. Enter the town, and the elder will greet you and raise the Lunar Whale for you. This ship will allow you to travel to the moon.

Lunar Whale
Inside of the Lunar Whale are some things of note. First, the top control area allows you to fly around. The crystal in the middle allows you to travel between Earth and the Moon. Below that is a sleeping area, which acts as a free inn. Finally, at the bottom is a Fat Chocobo, allowing you to access your storage at no cost.

Before you go to the Moon, make sure you have finished the Sylph Cave sidequest, because the reward is inaccessible after heading to the Moon. Additionally, if you wish to get Excalibur, now you have all the requirements to do so. With all that completed, activate the crystal and fly to the Moon.

The Moon
Despite all the dots on the map when you use Sight, there's currently only three places you can actually go. This is due to the Lunar Whale not being able to land on the crater-filled surface, only on flat areas. First, fly east to the Crystal Palace and then head south towards an isolated cave and land.

Hummingway Abode
There's not much of interest here, aside from being able to buy Ethers, Elixirs, and a new item called the Gysahl Whistle, which allows you to summon the Fat Chocobo anywhere you want.

Back outside, there is a plateau that has a ramp downward and will let you enter the set of caves. It is to the upper left of the strange Crystal Palace. Land somewhere on the plateau, and enter the cave below.

Lunar Underground


Pass through this cave, going east at the fork for a chest with a Golden Apple, guarded by 2 Eukaryote and 2 Prokaryote, then head north and make another east at the second fork for two chests with a Stardust and a Lunar Curtain. Exit to the north, then keep walking east to enter another cave. This one is empty, so just head south to exit. Once you exit, simply head west and enter the Crystal Palace.

Fusoya


When you enter the Crystal Palace, on the left will be a tile that will restore HP, and to the right is a tile that will restore MP. Go straight ahead, and a sequence will begin which will result in Fusoya joining your party. Fusoya is basically a beefed up Tellah, capable of using all White and Black magic, as well as Flare, Holy, and Meteor.

Return to Earth
Exit the lair, and return through the caves to the Lunar Whale.

Before touching the crystal, make sure you are completely prepared for a dungeon. Head back to Hummingway's Abode if you need more healing items. Once you return to earth you will be forced into the Giant of Babil, and will not be able to leave until it is destroyed. If you wish to do so, you may complete the Bahamut sidequest before you head back.