there are 12 kinds of fruits available in new leaf.

there are 5 possible town fruits:

6 kinds of island fruit:

and 1 special fruit:

there are a few ways you can get the fruits not native to your town.

before you have 100% town rating, ask isabelle for advice on increasing your rating. one of the quests involves you going to the beach. if you bring her back a seashell, she will reward you with a basket of non-native fruit.

if your friendship gets high enough with your villagers, they may give you fruit. you can also send villagers letters asking to swap fruits and attach a stack of fruit.

all of the fruits except cherries are used in recipes for harvest day. villagers will trade you the various ingredients needed (fruits, in this case) in exchange for a specific fish.

your mother can send you fruit all throughout the year. having 4 characters raises the likelihood as they all receive letters and can get different fruit.

you can purchase bamboo and persimmons at the fully upgraded nookling store.

obviously pertaining to the island fruits, but you can collect the fruits both from the beach outside and on tours.