Board the Lunar Whale, then examine the crystal to go to the Moon.
Again, you are under-leveled, I suggest you level up a little bit before you continue.

Fly the ship to the middle area and find stairs leading to a cave.
Board your ship nearby, then follow the map below.
You have to go through two tunnels, Western Lunar Path and Eastern Lunar Path, before arriving at the Crystal Palace.

After FuSoYa joins the party, make sure that you have the following formation:

• Front Row: Cecil, Edge
• Back Row: FuSoYa, Rosa, Rydia

Then backtrack to the Lunar Whale.

Bestiary Checklist for both Lunar Paths:

• [FFIVPR-060]060. Purple Bavarois - Western Lunar Path
• [FFIVPR-086]086. White Mousse - Western Lunar Path
• [FFIVPR-092]092. Balloon - Western Lunar Path
• [FFIVPR-128]128. Eukaryote
• [FFIVPR-130]130. Prokaryote
• [FFIVPR-131]131. Abyss Worm
• [FFIVPR-132]132. Black Flan
• [FFIVPR-133]133. Dark Grenade

Chests/Items:

• C-296: Golden Apple (Monster) - 1st chest.
  Monster: Eukaryote (x2), Prokaryote (x2)
• C-297: Lunar Curtain - 2nd chest.
• C-298: Stardust - 3rd chest.