Complex sentence: Alexander and his family moved to Miami, Florida, in 1961 and he went to New York in 1962 and started to play at Jilly Rizzo's jazz club Jilly's.
Split into simpler sentences: Alexander and his family moved to Miami, Florida, in 1961. He went to New York City in 1962, where he started to play at Jilly Rizzo's jazz club Jilly's.