You can purchase Nook Miles Tickets from the Nook Stop terminal for 2000 miles. Use it at your airport to fly to a random island, where you can gather materials and meet new residents.
You can redeem Nook Miles Tickets at your airport to fly to random islands. At these islands, you can find different types of fruit trees, bamboo trees, bugs, and fish. These islands each have a random animal resident. This page explains how it works.
How to Purchase a Nook Miles Ticket
Each Nook Miles Ticket costs 2000 Nook Miles, and can be purchased by using the Nook Stop Terminal in the Resident Services Tent, in the "Redeem Nook Miles" menu.
Before you use a Nook Miles Ticket, be sure to have tools in your pockets. You can buy them on the island with Nook Miles, but I recommend bringing your own tools.
How to Use a Nook Miles Ticket
After you buy a Nook Miles Ticket, go to your airport and talk to Orville (the dodo behind the counter). Say "I wanna fly" and then choose "use Nook Miles Ticket". Orville will explain the details before you go.
Harvest from the Island
You are free to harvest everything that you find on the island. You can dig up trees to take back to your island, hit trees and rocks to gather materials, shake trees for a chance of finding furniture (with a risk of finding wasp nests), dig up flowers with a shovel to take them home and plant on your island, and so on.
Meet an Animal Resident
Each island has an animal resident that you can talk to. If you talk to them more than once, you can invite them to move onto your island.
Find a Message in a Bottle
Sometimes, you can find a message in a bottle on the beach of a Nook Miles Island, so be sure to check all the beaches there.
Find a Fossil
There is a chance of finding a buried fossil on Nook Miles Islands, so be sure to check for buried objects.
Most of the islands that you find will have your island's native fruit, but sometimes you will instead find an island with trees that have a different fruit, or bamboo trees instead of fruit trees. To take full advantage of this, you can dig up the trees and take them back to your island to plant them there.
There is a chance of finding an island with a smaller island in the middle that has several tarantulas walking around on it. The tarantulas will keep appearing after you catch them, so you can catch infinite tarantulas here. As scary as this sounds, tarantulas are worth a lot of bells, so you may wish to drop most of your tools and fill your inventory with tarantulas before leaving Tarantula Island.
Bell Rock Island
You might find an island that has a smaller island in the center. You can only reach the center island by going to the north side and breaking the rock at the top. (Eat a fruit or other food and then hit it with your shovel to break it.) Then use the vaulting pole from where you broke the rock, to reach the center island. The rocks in the center are all bell rocks. Stand with your back to the edge of the island and hit each rock eight times with your shovel to make bells come out.