"In" is grammatically correct in that sentence. But the more natural way is to remove "At" and use "went to": "First I went to the British Museum". If you say "At first I was in the British Museum", it sounds like something unexpected is about to happen. "At first I was in the British Museum, but suddenly I found myself on Mars!"

CORRECT: Last summer I went to London. I did a lot of sightseeing. First, I went to the British Museum. (Then, I went to the London Eye.)

The present perfect is often used to describe past events. For example: "I have been to the British museum four times." The present perfect is used to describe past events which did not take place at a specific time.

Last summer...is past time so you cannot use "have been to"..present perfect tense