Brady enjoyed an exemplary junior hockey career in the WHL with the Medicine Hat Tigers. After scoring 81 points in 72 games and registering over 100 penalty minutes in 1985–86, Brady was chosen third overall by the New Jersey Devils in the 1986 NHL Entry Draft. He was returned to junior, then turned pro with the AHL's Utica Devils in 1988–89. He made his NHL debut on October 5, 1989.