The first two sentences look normal to me. People sometimes prefer "into" when they are talking about moving from one area into another, but "in" is also fine.
I don't think there is anything grammatically wrong with "inside", but it looks unusual to me. I generally use "inside" to tell somebody about a position inside of something rather than a movement into something. By itself, "inside" often means "indoors", and it is common: I am going in/into the house. (normal) I am going inside the house. (inside is unusual and emphatic) I am going inside. (normal)