There are five fruits that can be native to a town, six fruits that can be found on the island, and one that can be bought from the department store. This page explains how to get all of the fruits.
Whenever a player creates an Animal Crossing: New Leaf town, one of the following five fruits is selected as the "native fruit" of that town. The native fruit is the type of fruit that is on the town's trees when the player first moves in.
If you want to get any of the other four types of fruits that are not native to your town, you can do so in the following ways:
Before you get a 100% approval rating, you can ask Isabelle for advice. Do the things that she mentions. When she talks about going to the beach to get seashells, bring her a seashell and she will give you a basket of non-native fruit.
If your friendship with a villager is at a high level, the villager might give you some fruit, and if you are lucky, it will be a type of fruit that you don't have. You might also get fruit in the mail from villagers that you are friends with.
Another way to get fruits is to exchange friend codes with other players and trade fruits with them.
The following six types of fruits can be found on the island.
Before you can get island fruits, you have to unlock access to the island. Read the going to the island guide to learn how to do that.
When you go to the island, you start in a wooden building where you can buy souvenirs, go on tours, sell things, and check the ATM. Go through the south exit of this building to find a small island. There should be either banana trees or coconut trees on the sand, and trees with one of the other types of island fruits on the grass in the middle.
In the wooden building where you first arrive on the island, speak to Leilani, who is standing behind the counter. She will tell you a list of the tours that you can go on today. To learn more about island tours, check out the Island Tours page.
Some tours require you to give Tortimer some fruit. If you are on a tour where this is NOT the case (a tour where gathering fruit is NOT the objective), then you can keep any fruit that you take from the trees during the tour. It will be in your inventory when you are done with the tour. The fruit on the trees during a tour is random, so just go on many tours until you have collected the types of fruit that you don't have. Send these back to your town using the box next to Leilani's counter.
Before you leave the island, be sure to put any fruits and items that you want to take home in the box to the right of Leilani, who is standing behind the counter in the island hut. When you go back to your town, the basket on the dock will have the items that you put in the island box.
Persimmons are sold in the department store. Read how to unlock the department store.