Snow is the son of former NFL Los Angeles Rams Pro Bowl wide receiver Jack Snow. Snow attended Los Alamitos High School in Los Alamitos, California and played baseball and football and basketball with future teammate Robb Nen. After high school, Snow played three seasons at the University of Arizona. He was drafted by the New York Yankees in the fifth round of the 1989 baseball amateur draft. Snow broke into the Majors with the Yankees at the end of the 1992 season.