Stevens played his junior hockey with the Kitchener Rangers, the same team as his brother Scott, and was selected 58th overall in the 1984 NHL Entry Draft by the Vancouver Canucks. While still playing in Kitchener in 1984–85, he received a surprise six-game callup to the Canucks, recording three assists.
However, he would have a difficult transition to pro hockey, struggling to produce in two seasons in Vancouver's farm system with the Fredericton Express of the American Hockey League. He did, however, make himself a physical presence, recording over 200 penalty minutes both years.