Sometime we waved trees, because fell down some plums and some plums fell down. Sometime we waved trees and some plums fell down.

Clarity mistake, make it clearer that the reason for shaking the trees was to make the plums fall

Homestay fourth day Homestay fourth day

Today was special day！ ! I went to an olympic sport center by bike. Today was special day! I went to an olympic sport center by bike.

There is two people ride on th-able bikes. There is two people ride-able bikes.

I ridode in the back seat.( (the front of ride was my host family's mother) . When we came back home way, we found some plum trees. I rode in the back seat (the front of ride was my host's mother). When we came back home way, we found some plum trees.

Past tense since this happened in the past. "ride" can be used as a verb and a noun so make sure you know when you use it effectively

My host's family made plum’s compote. My host's family made plum compote.

It was a little bit sour, but it’ was very delicious. It was a little bit sour, but it was very delicious.

Past tense, the first part of the sentence was correct but for some reason you switched the tenses in the seconds part of the sentence.

It’s a cherished memoriesy. It’s a cherished memory.

When using "a" you talk about a single object so instead of "memories" it would "memory"

I had anve other stories, but it wroteI will write them next time. I have other stories, but I will write them next time.

Past, present and future tense mix. Here using the word "had" means that you have stories but then you forgot it, use "have" instead since you still remember it. "Wrote" is the past tense but since you about to do the action, it should be future tense and should be "to write"