Suter was part of a hockey family. Bob Suter, one of three hockey-playing older brothers, was a defenseman on the "Miracle on Ice" United States team that won the gold medal at the 1980 Lake Placid Olympics. His father, Marlow Suter, helped establish the Madison Capitols minor hockey system in Wisconsin that all four of his sons played in.
After going undrafted in 1982 and 1983, Suter was selected by the Calgary Flames in the ninth round (No. 190) of the 1984 NHL Draft following his freshman season at the University of Wisconsin. He played one more season of college hockey before turning pro in 1985-86, when he became an instant hit, winning the Calder Trophy as the NHL's top rookie after scoring 18 goals and 68 points in 80 games.