Taylor was selected by the Los Angeles Kings in the 15th round (No. 210) of the 1975 NHL Draft after he scored 20 goals in 29 games in his second season for Clarkson University. He scored 26 goals in 1975-76, then set school records for goals (41) and points (108) in 1976-77 before turning pro with Fort Worth of the International Hockey League.

The seven games Taylor played in the IHL at the end of the 1976-77 season turned out to be the only ones he played in the minors during his career. He made the Kings roster out of training camp in the fall of 1977 and finished his first NHL season with 22 goals and 43 points.

Taylor more than doubled those numbers in 1978-79, when coach Bob Berry first put him together with center Marcel Dionne and left wing Charlie Simmer on what became known as the "Triple Crown Line," one of the most prolific NHL lines in the early 1980s. Taylor, the left wing, had NHL career highs in goals (47) and points (112) in 1980-81, when he was named an NHL Second-Team All-Star and was part of the first line in NHL history to include three 100-point scorers. He followed that with a 39-goal, 106-point season in 1981-82.

After missing sizeable parts of the next two seasons with injuries, Taylor was healthy in 1984-85 and rebounded by scoring 41 goals and finishing with 92 points. He scored his 400th NHL goal on Dec. 29, 1990 and reached the 1,000 points with an assist on Feb. 5, 1991. At the end of that season, he won the Masterton Trophy for perseverance and dedication to hockey, as well as the King Clancy Trophy for leadership and humanitarian efforts.

Taylor reached the Stanley Cup Final for the only time in his career with the Kings in 1993. He scored three goals and eight points in 22 postseason games, although the Kings lost to the Montreal Canadiens. He retired after the 1993-94 season with 1,069 points (431 goals, 638 assists) in 1,111 NHL games, as well as 59 points (26 goals, 33 assists) in 92 Stanley Cup Playoff games. His No. 18 was retired by the Kings on April 3, 1995.

The Kings hired Taylor as general manager in 1997, and he later worked for the Dallas Stars and St. Louis Blues. In 2019, Taylor earned the Stanley Cup ring he never won as a player when the Blues defeated the Boston Bruins in Game 7 of the Final for the first championship in their history.