Treasure Isle is unlike the other side-quests since you can do it the moment you get Grind from Lemuria (although it's a lot easier if you do the majority of Mars Lighthouse first). It also has a lot of extra equipment and items.
Finding Treasure Isle
The island surrounded by rocky spires to the north of the Eastern Sea and Izumo is Treasure Isle - you need to go to the north of it then follow the path in-between the rocks around to the shore, then disembark and go in the cave.
Inside Treasure Isle
To begin with, most of the chests are already empty - someone has already been here. However, you can check the left-most door in the first room, and the chests inside contain some coins and a Lucky Medal. Back in the first room, keep going up into the next area.
The six chests on your left are all empty, so just hop straight up to the button. This will make rocks shoot up from where the brown circular stones were. Use Grind on the right-hand one on the platform you are standing on (using Grind on the rock above leads nowhere). Hop to the right and down through the door at the bottom. In the next room, go right and around the bottom of the enclosure, then around the stepping stones. Move the pillar to the square directly below the button, then hop back around to the other side of the enclosure. Use Move to push the pillar onto the button, and this will raise a rock that allows you to jump across to the door on the right.
In the next room, use Grind on the right-hand rock and continue up across the stones. Up the ladder and across the tightrope you can hop across the rocks and get a chest containing the Jester's Armlet. Equip this on whoever you want. Return back the way you came, and go left to the pillar. Use Move, then push it yourself into the groove, then hop up to the top door.
Go down at the first fork, then take the first left and confront the Mimic for some Power Bread. Head down to the bottom door and Move the pillar across into the groove. You can now climb up the ladder and hop across the rock to get the six chests on the left, which are (in no particular order): a Slyph Feather, a Rusty Axe, some Stardust, a Cookie, a Psy Crystal and 911 coins. Now, hop back across the rock and return to the room you came from. Go right, back to the fork and go up, hugging the top wall until you get to the stairs down.
Use Lift on the rock, and go around to the left. The chest here contains the Iris Robe, so equip that on whoever. Go down, and then left. Use Lift to proceed past the rock to the door. The next room is quite big. Use Lift on the center boulder and then climb up the ladder. Hop over the rock and around the tightrope, then across another to confront the Jupiter Djinni. Beat Gale in a battle and it'll join you. Now, go back up, then hop across to the right-most tightrope. Go down this one and grab the Flame Brand from the chest - equip this on whoever has the lowest Attack.
Now leave the room and re-enter it to reset the boulders. Lift the left-hand ones this time, then the back right one. Go under the back right one and hop across to the ladder. Go up, then down the left-hand tightrope. At the bottom, hop all the way across to the right and go down the ladder. Ignore the bottom door, just go through the top one. The left-hand path doesn't lead anywhere, so go right and up through the door.
Save before you go any further, and heal up. Once you have, hop across and go up the stairs to confront the Star Magician!
Boss Battle: Star Magician and Balls
Valukar had the ability to use your own summons. Sentinel was immune to Psynergy. The Star Magician has an army of balls which each do different things based on the colour.
The Guardian Balls are the green ones. These can use Guard Aura on the Star Magician to greatly reduce the damage he takes during that turn. If you're going to use summons it is a good idea to make sure there are no Guardian Balls around. Other than that, they will just defend.
Refresh Balls are blue. These may not do anything for a turn, but will occasionally use a move called Earnest Ply on the Star Magician to heal him for a little over 1100 HP. So, obviously, these are the ones you want to get rid of before any others.
Anger Balls are red. These will attack, and can also do a powerful suicide move called Angry Mine too, so watch out.
Thunder Balls are purple, and will use an assortment of relatively weak Jupiter Psynergies on you like Shine Plasma and Flash Bolt.
The Star Magician is obviously the strongest of them all. He attacks twice a turn and will use a selection of powerful Jupiter and Mercury Psynergies like Spark Plasma and Megacool. He can also use an attack called Mine Ball which hurts quite a bit, as well as summon more balls to help him out. With regards to balls, make sure you kill off the Refresh ones first, then the Guardian ones. The Angry balls have a tendency to suicide themselves, and the Thunder ones don't pose much of a threat.
Keep everyone well-healed and reel off attack Djinn and Psynergies to dispatch Guardian and Refresh balls and hurt the Star Magician. Summons, when you get them, will help clear out some of the other balls as well as do a good amount of damage to the Star Magician. Hiding behind barrier Djinn helps too. Either way, this battle will take quite a while.
Due to all the balls you killed off during the battle, you'll get an enormous amount of coins and experience afterwards, but the real prize is waiting for you. Walk up and claim Azul! It requires 3 Venus and 4 Mercury Djinn to summon. Now, Retreat from Treasure Isle.