Robert Stevenson started his football career at Third Lanark in the Scottish League (1892-93). The following season he played in the Scottish Cup semi-final against Celtic. Third Lanark lost 5-3. He moved to London to work for Donald Currie & Company and played for the works team Old Castle Swifts. He also occasionally turned out for Woolwich Arsenal. In 1895 Arnold Hills, the owner of the Thames Iron Works, decided to establish his own football team. During the club's first season Stevenson was captain of Thames Iron Works and played as full-back and centre forward. At the end of the season he went back to Scotland to play for Arthurlie.