The correct version is:
Rachel Miller has been dancing since the age of four; she's also a gifted singer. People say that a talent for the musical and performance arts runs in the Miller family, starting with Hannah Miller, Rachel's great grandmother. However, Rachel's younger sister, Janet, isn't too fond of singing and dancing; she prefers losing herself in a good book or movie.
[1] is a semicolon - In this sentence semicolon has been used to join two independent clauses.
[2] is a comma - We used comma to explain Hannah Miller who is Rachel's great grandmother.
[3] is a comma - We put comma after however when the sentence begins with this word.
[4] is a semicolon - We used the semicolon to join two independent clauses.
