This page explains how to go to the island resort. Tortimer will tell you about it, then Kapp'n can take you there.
Step 1: Upgrade to a House
When you first start the game, you live in a tent. Pay off your loan to upgrade to a house. Next, you have to talk to Tom Nook to get your next home loan, then pay it off with the ABD machine in the post office. You have to pay off this first house loan before you can go to the island resort.
Step 2: Talk to Tortimer
After you pay off the loan mentioned in the previous step, the next time you play, Isabelle will introduce Tortimer at the start-up screen. After this, go down to the dock on your beach to find Tortimer standing there. Talk to him and he'll tell you about the island. Note! If you forget to talk to Tortimer, just pay off another house loan to make him show up again.
Step 3: Wait a Day
After you talk to Tortimer on your dock, you have to wait until the next day. Then, go to the dock again. Kapp'n will be there in a boat.
Step 4: Pay Kapp'n to Take You to the Island
To go to the island resort, pay Kapp'n 1000 bells. Then you will hop into the boat and he will take you to the island.
You Can Skip the Song
If you don't feel like waiting through Kapp'n's sea shanty that he sings on the way to (and from) the island, press the B button repeatedly until he stops.
Things to do on the Island
When you are on the island, you can go on tours, collect fruit, and catch bugs and fish that might not be available in your village. Leila, the little girl Kappa, will buy things from you at 5% of their Re-Tail selling price. You can win medals by going on tours, then use the medals to buy things from Grams's souvenir shop. Sometimes she sells a membership to Club Tortimer, which allows you to go to the island with a random player over the Internet.
Sending Things Home with the Box
To send things back to your town from the Island, open the box to the right of Leilani's counter (near the top left). Anything that you put in the box is sent to the basket on the dock in your village. Note! If you have anything in your pockets when you leave (except tools borrowed from Lloid), they will automatically be sold to Leila (at 5% of their value).