Once you’ve bought your house and completed the first upgrade (i.e. paid off the 39800 bell loan), Isabelle will tell you that Tortimer, the former mayor, has come to visit.
You’ll find him on the dock on the beach of your town. Tortimer will tell you about his island which you can visit from the next day by speaking to Kapp’n from the dock, at 1000 bells per trip.
On arrival, you’ll be in an exotic shack on the island. Here, you can register for tours, purchase from the gift shop or barter with lil’ Leila.
Anything you can grab from the island, you can sell to Leila but for a very low value– don’t bother doing this unless you have to!
Instead, put everything of value in the basket next to the front desk and you can pick them up when you return to town, from the basket by the dock.
When you step outside the shack and onto the island, there is a friendly gyroid next to the door who will loan you equipment.
You can the net to catch bugs, fishing rod to fish, axe to chop trees and shovel to plant some of the tropical fruit on the island. You can even use the diving suit to dive into the water.
Use your borrowed gear to catch beetles and sharks as they are the most valuable animals on the island. You can then sell them back on the mainland for a healthy profit.