Dwyer's childhood dream was to become a professional footballer and his career began to progress at the age of 14 when he was picked to represent the Cardiff Schools side for the first time. He gained international recognition soon after when he was selected to represent Wales at under-15 level in the Victory Shield, a youth competition held between the Home Nations. Despite this, he initially struggled to attract interest from clubs until he was invited to train with the youth team at Cardiff City by Harry Parsons, a long serving employee of the club. Dwyer began training with the side every Tuesday but still held little hope of fulfilling his dream. He was due to start a carpentry apprenticeship with a company at Cardiff Docks which had been arranged by his brother-in-law who worked at the site. To Dwyer's surprise, he was offered a contract by Cardiff when he turned 16 in 1969, earning around £5 per week.