Adams was born in Stoke-on-Trent and began his career with local side Stoke City making his professional debut against Coventry City in the Full Members Cup in 1985–86. He made a huge impression with his performances for Stoke in his debut season, with his attacking creativity and deadly accurate crossing ability earning him a call up to the England U21 side for their European Championship semi-final tie in Italy. With Stoke's finances poor and many top-flight sides showing an interest in signing him, he was sold to Everton in June 1986 for a fee of £150,000 after the Toffees beat off competition from Arsenal and Tottenham Hotspur, much to the annoyance to the Stoke supporters who feared that their club was becoming a 'selling club'.