All of us had been learning English and we were members in the same English study group.

NOTE: You can safely delete the word "we" in this sentence.

OPTION A: I had always believed we were good friends, but actually that was wrosomething was wrong.¶
OPTION B: I had always believed we were good friends, but trouble was brewing.

One of them H started being group members became very selfish during the trip.

I washad been aware that Hshe was gradually actingbecoming more selfish, but I didn't really mind because I though she was just gettingmerely tired.

NOTE: You can delete the 2nd "I" from this sentence.

I knowThis can easily can happy if you walk around all day long you will getand feel worn out.

However, one of my friends N blew up on the third day.

N was likeShe said, "I can't stand any morebeing with her, if she annoys us tomorrow, I might give her a hard time. any more. If she continues to bug us tomorrow, I will tell her exactly what I think!"

(This story has awill be continuation, so I'm going to write about it agained in my next entry.)

I had always believed we were good friends, but actually thatI was wrong.

"But actually that was wrong" is understandable, but I feel like I would say something more like, "but actually, I was wrong" or "but actually, I turned out to be wrong." English speakers tend to say "I" was wrong or "you" were wrong, and if a belief is mistaken, there's a focus on the person being wrong. "That was wrong" isn't really said so often in English compared to Japanese (違います/違いました).

One of them, H, started being selfish during the trip.

I was aware that H was gradually acting selfish, but I didn't really mind because I thought she was just getting tired.

However, one of mymy other friends, N, blew up on the third day.

N was like, "I can't stand her any more with her, if she. If she keeps annoysing us tomorrow, I might give her a hard time."

I didn't know she felt the same way as me, but I said that maybe we shouldn't make a fuss until we cameget back to Japan.

"Get back" or "go back" sounds more natural to me than "come back." Also, I think the verb should be written in the present tense and not the past tense because you're writing about what you said at that moment: "Maybe we shouldn't make a fuss until we get back to Japan." At that moment, it was the present, not the past.

This story has a continuationere's more to this story, so I'm going to write about it again in my next entry.

"The story has a continuation" sounds awkward in English to me, even though it sounds natural in Japanese (続きがあります). Native English speakers would say something more like "There's more to this story" or "There's more that happened afterward."

One of them, we'll call her H, started being selfish during the trip.

I was aware that H was gradually acting selfish, but I didn't really mind because I thought she was just getting tired.

However, one of my friends, N, blew up on the third day.

N was like, "I can't stand her any more with her, i! If she annoys us tomorrow, I might give her a hard time."

I didn't know she felt the same way as me, but I said that maybe we shouldn't make a fuss until we camebefore we go back to Japan.

All of us had been learning English (for a while) and we were members in the same English study group.

One of them, H, started being selfish during the trip.

I know that if you walk around all day long, you will get worn out.

However, one of my other friends, N, blew up on the third day.

N was like, "I can't stand her any more with her, i. If she annoys us tomorrow, I might give her a hard time."

This story has a continuation, so I'm going to write about it againI will continue this story in my next entry.

I had always believedthought we were good friends, but actually that wasthat turned out to be wrong.

One of them Hー H ー started being selfish during the trip.

I was aware that H was gradually acting (more-and-more) selfish, but I didn't really mind because I though she was just getting tired.

N was like "I can't stand any more with her, iher (behaviour/whining/demands) any more. If she annoys us tomorrow, I might give her a hard time."

I didn't know she felt the same way as me, but I said that maybe we shouldn't make a fuss until we caome/get back to Japan.

She agreed with my idea and we came back to Japan without any trouble.

Or:
we came back to Japan without an incident.
we came back to Japan without a fight.

I had always believed we were good friends, but actually, that was wrong.

One of them, H, started being selfish during the trip.

I was aware that H was gradually acting selfish, but I didn't really mind because I thought she was just getting tired.

I know if you walk around all day long you will get worn out.

However, one of my friends, N, blew up on the third day.

N was like, "I can't stand any more withof her, i. If she annoys us tomorrow, I might give her a hard time."

“N was like…” is a common casual spoken construction, but more formally you’d want “N said,” or similar.

I didn't know she felt the same way as meI did, but I said that maybe we shouldn't make a fuss until we came backreturned to Japan.

“Me” would be clearly understood, but is technically not right. Me is an object pronoun while I is a subject pronoun.

“Came back” would be understood, but is a little awkward. “Come” is a verb where the object moves toward the speaker. In this instance, you’re in the UK. So while Japan is “home,” you can’t “come” there from abroad, you’d “go there” or “return there” etc.

She agreed with my idea and we came backreturned to Japan without any trouble.