Born in Glasgow and raised in the city's Drumchapel area, McGrain played for Queen's Park Strollers and also featured for the Scotland Schools team, where he was scouted and signed for Celtic in May 1967, aged 17. He became one of the so-called 'Quality Street Gang', the great Celtic reserve team that also included players such as Kenny Dalglish, Lou Macari, Davie Hay and George Connelly, who eventually took the places of the ageing Lisbon Lions. At first, McGrain was regarded as a midfielder but was utilised in a variety of roles in the reserve side before becoming established as a right-back. He had a spell at Junior club Maryhill to gain experience playing with adults, and also made a fruitless attempt at a college course in engineering prior to becoming a full-time professional.