Young gained attention as a star hockey player while still in high school. He played his prep school hockey at St. Mark's School in Southborough, Massachusetts, playing with teammates that included fellow future-NHL players Doug and Greg Brown. His play allowed him to play with the United States in the World Junior Ice Hockey Championship in 1985, the beginning of a long international career representing the United States. Following the championship Young went to play for the Boston University Terriers. He played there 2 years, winning rookie of the year honors in 1986. Young was also drafted by the Hartford Whalers in the 1st round, 11th overall in the 1986 NHL Entry Draft while still in college. For the 1987–88 season Young spent the majority of the season with the U.S. National team. In addition to playing on the national team Young played in the 1988 Winter Olympics and made his debut with the Whalers, playing in 7 games. The following season he played full time with the Whalers, scoring 59 points in 79 games. Young played one more full season with the Whalers and played half of the 1990–91 season with the Whalers before being traded to the Pittsburgh Penguins. Young helped the Penguins win the Stanley Cup in 1991. For the 1991–92 season Young spent the majority of the year playing in Italy in addition to a brief stint with the U.S. National Team and representing the U.S. in the 1992 Winter Olympics. Prior to 1992–93 season Young returned to the NHL and was traded by the Penguins to the Quebec Nordiques. He played 3 seasons with the Nordiques and remained on the team when they moved to Colorado and became the Colorado Avalanche. He played 2 seasons with the Avalanche and won his second Stanley Cup with the Avalanche in 1996. Prior to the 1997–98 season Young was traded to the Mighty Ducks of Anaheim and played one season with the Ducks. The next offseason Young signed with the St. Louis Blues. In his first stint with the Blues he played 4 seasons, enjoying the best season of his NHL career during the 2000–01 season, scoring 73 points and scoring 40 goals, both career highs and the only time Young eclipsed 40 goals in a season. Young also represented the U.S. once again in the 2002 Winter Olympics. Prior to the 2002–03 season Young signed with the Dallas Stars and played 2 years with them. Following the cancelled 2004–05 season lockout Young rejoined the Blues for the 2005–06 campaign. While the Blues finished last in the league that year, Young proved that he was still a strong hockey player, leading the team with 49 points. Following the season, Young retired from hockey. Young finished his career with 1181 career NHL games, 342 goals and 414 assists for 756 points. In 2011, Young returned to St. Mark's School, this time as the coach of the Boys' Varsity Hockey team for which he once played. As a coach, Young has restored credibility to the program with three winning seasons, two Barber Tournament championships and two Boys' Holiday Showcase championships.
