To be able to get to Bamboo Island, you have to go through the Eastern Pyramid of Aegis with Felio's Emblem. There is a room with a bunch of coffins, and if you head south between those coffins, you reach a room with a bed and a bath. There is a coffin below the bed and bath, and if you walk under that coffin and head south, you find a secret exit from the pyramid where a red bird flies past. Tap the bird with the stylus, then press A when the map floats by. Bamboo Island will be added to the SeaNavi.
On Bamboo Island, go into the house and talk to the man in the bed. He comes from a long line of thieves. He is sick and dying. Agree to let him pass his thieving skills to you.
Find the Amsonia
Your first trial is to find a medicine called Amsonia in Caldoaxa. You will find this medicine in the room of the Spice Shop that Ike got locked into with the silver beetles. You need to go back to that room by shoving Ike out of the way (try walking up to him from an angle,) then get the Cloves from the top-right shelf. Now you can return to Bamboo Island and talk to the old guy again.
Deliver the Letter
You have to deliver the letter to the woman who told her daughter not to speak to strangers. She's standing by the palm tree on the east side of town. She will ask you to tell the thief some things. Go back to Bamboo Island and talk to the the old man.
Get the Thief Costume
Now check on the treasure chest. The boy will rush over and show you how to pick the lock. The old man will pass away just as the boy opens the chest. The boy will be convinced to be a thief when he grows up, but he lets you have the Shadow Thief costume. Go over and press A to get it, then change into it.
Talk to the boy, then go outside and find the hidden path that is to the east of the house. You will find the old man's grave. You might see a UFO here. If you do, blow into the microphone until the UFO lands. The alien will give you the UFO decal. It will make all enemies on the screen disappear. You have a random chance of encountering the UFO when you visit the old man's grave, so if you don't see the UFO now, try again at other times.
Next, you can go to Ft. Eagle to get the Fisher King outfit.
Also in Ft. Eagle, you can make good use of your Thief outfit. If you don't have a Thief Dagger, go back to Aegis and kill Sarcophagi until you get it. Then return to Ft. Eagle with the Thief Dagger equipped, and make your way to the room in the Facility where there was a locked treasure chest.