The Uruguay forward was able to train with Barca on Friday morning after the Court of Arbitration for Sport relaxed the "excessive" elements of the four-month ban incurred for biting Italy defender Giorgio Chiellini during the World Cup.
"I'm very happy to feel like a footballer again and to be with my team-mates," Suarez told the club's official website.
"I was really looking forward to starting. The situation made me feel very uncomfortable. I'm paying for an error I committed and have apologised for, but it needs to be forgotten about. Now I have to think about the future, which is Barcelona, the club I dreamed about playing for."
Suarez completed a £75million move to the Catalans in July but had been forced to train away from his team-mates due to the ban.
The 27-year-old failed in his appeal to have the entire suspension - covering "all football-related activity" - removed, but was permitted to train and play in friendlies for club and country after the CAS ruled the punishment was overly harsh.
Suarez will be presented in front of Barca fans at the Nou Camp before Monday's friendly against Mexican side Leon, when he could make his debut for the Blaugrana.
However, he will have to wait until the weekend of October 25 to make his competitive debut - the weekend of the first Clasico of the 2014-15 Primera Division campaign against arch-rivals Real Madrid at the Bernabeu.
"Now I have to wait to play again, to meet the fans and feel what it's like to play at the Nou Camp," Suarez added.
"I'm very grateful to the club for everything they have done for me in the past few weeks and days. They have treated me very well and I'm grateful to them because I've completed a dream, which is to play for Barcelona."
The CAS felt Suarez's ban broadly fitted the crime - having "committed an act of assault" - but believed extending it beyond the pitch was excessive, even taking into account it was the third time he had been found guilty of biting an opponent after incidents with PSV Eindhoven's Otman Bakkal and Chelsea's Branislav Ivanovic.
Brazil forward Neymar also trained with the main group on Friday, the club confirmed on their website, despite not having been given the medical all-clear after breaking a bone in his back during Brazil's 2-1 World Cup quarter-final win over Colombia.
Xavi, who has agreed to play on for at least one more season despite being linked with a move to either Qatar or the United States over the summer, was also involved, while former Arsenal defender Thomas Vermaelen continued with specific physio work.
Barca also confirmed, however, that new signing Marc-Andre ter Stegen had been ruled out for three weeks with a back injury, meaning Chile international Claudio Bravo looked set to start in goal for the league opener at home to Elche on August 24.
- Sports & Recreation