Beamish started his career with Fourth Division Tranmere Rovers in 1966, under the management of Dave Russell. Rovers missed out on promotion at the end of 1965–66 after finishing behind fourth place Colchester United on goal average. Promotion was achieved in 1966–67, after the club finished fourth, four points ahead of Crewe Alexandra. Tranmere retained their Third Division status in 1967–68, 1968–69, 1969–70, and 1970–71. Beamish was the club's top scorer two out of his six seasons with the club. Now managed by Jackie Wright, Rovers struggled in 1971–72, and in March 1972, Beamish switched to league rivals Brighton & Hove Albion.