Barnes was drafted fourth overall by the Winnipeg Jets in the 1989 NHL Entry Draft. On November 25, 1993, the Jets traded him along with a sixth round selection (previously acquired from the St. Louis Blues; Chris Kibermanis) in 1994 to the Florida Panthers for Randy Gilhen. In Florida, he was among the leaders on the teams, who helped carry the Panthers to the 1996 Stanley Cup Finals, facing the Colorado Avalanche. Then on November 19, 1996, the Pittsburgh Penguins traded Chris Wells to the Panthers for Barnes and Jason Woolley. The trade to the Penguins has been considered the worst in Panthers history.