Fulton joined Celtic from their Boys Club in 1987 and made his league debut during the 1988-89 season. After his impressive display in a Scottish Cup semi-final defeat of Hibernian that year he was likened by his manager Billy McNeill to Italian forward Roberto Baggio, and the Baggio moniker stuck with him throughout his career. He had become a first-team regular by 1990 and helped the side to the 1990-91 League Cup final, where they suffered a 2-1 defeat to Old Firm rivals Rangers. Unusually for a player who spent six seasons with Celtic, he was destined not to earn a medal with the Parkhead club.
Fulton left Celtic Park in August 1993, Bolton Wanderers paying £225,000 for his signature. However, he fell from favour within months of his move to Burnden Park and spent part of the 1993-94 season on loan to Peterborough United before returning to Scotland at its conclusion, in a £100,000 move to Falkirk. The Bairns enjoyed a profitable first season back in the Premier Division, challenging for European qualification and eventually finishing fifth, only six points behind 2nd place Motherwell. Fulton re-established himself as a first-team player, playing 33 times during the campaign.
In July 1995, Jim Jeffries resigned as Falkirk manager to take-over at Heart of Midlothian and Fulton was one of the first players he signed for the Edinburgh club. The side had been near the bottom of the table at the time Fulton signed but recovered to finish 4th and reach the Scottish Cup Final, where Rangers proved too strong for them. He helped Hearts to the League Cup Final the following season and scored their first goal in the Celtic Park showpiece. Rangers were again the opposition and once again they had the measure of Hearts, two individual goals from Paul Gascoigne turning the match in the Glasgow side's favour.
In 1997-98, Hearts mounted an unexpected challenge for the league title, with Fulton in the vanguard. He was the only Hearts player to start every league game and his six goals from midfield led the side to an eventual third place finish. They also reached the Scottish Cup Final for the second time in three years and ended a 36-year wait for silverware by defeating Rangers 2-1. Due to an injury to regular skipper Gary Locke, Fulton was appointed captain for the occasion and he set his side on the way to victory after he was brought down in the box to win a first minute penalty kick from which they took the lead.
Fulton earned selection to the Scotland national team squad for the first time in September 1998, for a World Cup qualifying game against Lithuania. His Hearts team-mate Neil McCann made his debut in that game but Fulton remained on the bench. Injuries and a loss of form ensured he wasn't selected in the next squad and as a result, never earned a full international cap.
After 1998, Fulton struggled to match the form he had achieved in his first three seasons with Hearts, the aforementioned injury concerns compounded by his difficulty in maintaining his optimum playing weight. By the 2000-01 season he was no longer an automatic selection in midfield, although he did occasionally play as an emergency left back. In May 2001 he was informed by new Hearts manager Craig Levein that he was free to leave the club and, after intermittent appearances during the 2001-02 season, Fulton left Tynecastle when his contract expired in May 2002.
Jim Jeffries was by this stage Kilmarnock manager and he again moved to sign Fulton, offering him a two-year contract with the Ayrshire club. Fulton moved to Partick Thistle in 2004 but was sacked by Jags manager Dick Campbell in April 2005 for a breach of club discipline. Fulton and team-mates Andy Dowie and Jamie Mitchell had left Raith Rovers' Starks Park for a local pub after learning they had not been selected in the Partick side, rather than watch their club's game. Mitchell's plea that the players did not consume alcohol and were only discussing problems in his personal life in a private setting was disregarded by Campbell.
Fulton is currently a coach at former side Hearts, in charge of the club's Under-13 side.