In my town, the only fruits I have on the trees are peaches. I've heard of getting other fruits, such as an apple or an orange. How can I get those fruits?
The only way to get other fruits are to be given them by another player or villager. As far as I can tell there's no way to force a villager into giving you fruit as a gift, but there's a random chance they might give you one in exchange for doing a task for them. Due to this the easiest and most reliable way to collect all the fruit is by trading with other players.
Once you do acquire the fruit you can then plant it in your own town, and it will grow into a fruit tree. Fruits not native to your town will sell for more money, so they're a good way to make a bit of extra cash. Only native fruit trees have a chance to grow perfect fruit though, and any perfect non-native fruit planted in your town will just grow a regular fruit true.
You can also get fruit off Tortimer's island, but it'll only be Exotic Fruit, not the regular fruit you'd find in town.
You can get other fruits either by visiting your friends' towns, and taking whatever they have on your trees and planting it in your own town, or taking the fruit from the island and planting that in your own town, or by sometimes getting random fruit as rewards for helping out your villagers, and then planting those.
A note, though - fruits like bananas that grow on palm trees can only be planted on your beaches, and other trees can only be planted off the beach.
Once your town's Nookling store is upgraded to the TNT Emporium, you can buy persimmons in the gardening section. Along with Bamboo.
You can do tasks for isabelle or go to the island and collect fruit from there, also get some from other players & you may also get a fruit as a reward from villagers after doing a task for them. :) hope this helped!
When you go on tours on the island, just pick some fruit from the fruit trees. When you are done with the tour it will still be in your possession.