My native fruit is cherries. I have spent many nook miles trying to get different islands to collect fruits and bamboo. I have only been able to get oranges and obviously palm trees. I keep landing on islands with more cherries and it’s incredibly frustrating. Is there a specific time of day or something I need to be flying to get different stuff??

the islands will only have your native fruit and your "sister fruit", which for you seems to be oranges. the same goes for flowers, they'll only have native and your sister flower

here's a guide that explains the details of all the different islands if you're interested

As far as I've been able to tell, having used well over 100 NMTs, you can only get islands with your native fruit or one other fruit. For me, my native is peaches, and my non-native is pears. I don't believe it's possible to get other fruit through any other means besides visiting other players' islands.

On that note, if you want to do a bit of a fruit exchange, hit me up.

How do I get bamboo then??

Bamboo can come from NMT islands - it's counted as a tree, not a fruit. Just need to get lucky, I guess.