Born in Boston, Lincolnshire, Garner was educated in his hometown at Boston Grammar School. He played for local club Boston United before joining Blackburn Rovers as an apprentice, turning professional in July 1978. His debut came the following season at Newcastle United in the Second Division, the first of 474 league games for Rovers, in which he scored a club-record 168 league goals. Rovers were relegated in Garner's first season as a senior player, but he helped them win promotion at the first attempt, and they only narrowly missed out on a second successive promotion in 1981.