He started his career at Tranmere Rovers in 1965, and in six years at the club would make close to 200 appearances. From March 1972, he then spent three years with Brighton & Hove Albion, making around 100 appearances. In May 1974 he signed with Blackburn Rovers for £25,000, and went on to play close to 100 games with the club in a two-year spell. Bought by Port Vale for £12,000 in September 1976, he was voted the club's Player of the Year in 1977–78, and was the club's top scorer for two consecutive seasons. Sold to Bury for £35,000 in September 1978, he played around 50 games, before he moved back to Tranmere Rovers in 1979. In 1981, he transferred to Swindon Town, before he announced his retirement the following year. He later served as Swindon's manager in the 1983–84 campaign.
Beamish started his career with Fourth Division Tranmere Rovers in 1965, 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.
Beamish enjoyed a successful start to his Brighton career, as the club won promotion as Third Division runners-up in 1971–72. However the club struggled in the Second Division, and finished in last place in 1972–73. Manager Pat Saward was replaced by Brian Clough, but the 1973–74 campaign was also difficult, as Brighton avoided their second successive relegation by eight points.
Beamish was signed by Gordon Lee's Blackburn Rovers for a £25,000 fee in May 1974. Whilst at Blackburn, he endeared himself to the fans, who would chant "Beamo" whenever Beamish's shot at goal flew over the bar. To Blackburn fans of a certain generation, a 'Beamo' is any off-target shot. The 1974–75 season was successful, as Blackburn were promoted to the Second Division as champions of the Third Division. Rovers finished mid-table in 1975–76 under the stewardship of Jim Smith.
He joined Third Division Port Vale for £12,000 in September 1976 and went straight into the first team. He became the club's top scorer in the 1976–77 season with 18 goals in 44 games; three of these goals came against Grimsby Town at Blundell Park on 3 January. He was voted Player of the Year for his 16 goals in 49 games in the 1977–78 season; he scored ten goals more than his closest rival Jeff Hemmerman, as the club were relegated into the Fourth Division. He started the 1978–79 campaign with four goals in four consecutive games, before he was sold on to Third Division Bury for £35,000 in September 1978.