Bellamy was frozen out by Mancini at Eastlands but decided against moves to other Premier League clubs, and Celtic, to drop down to the Championship to sign for his home-town club.
The 31-year-old revealed last week that he wouldn't be included in City's 25-man Premier League squad, sparking a scramble for his signature, but he told a press conference he had no problem with his parent club.
Bellamy said: "Nothing's difficult about this. I had the greatest time at Manchester City at a great period in their history. I was bought to take the club on to the next stage and I feel I played a big part in doing that.
"The players they're signing are incredible. The next couple of years I'm sure they'll get the Premier League title, and I hope they do.''
Bellamy refused to comment on reports City would not allow him to join a rival Premier League club but praised manager Roberto Mancini. "There's no ill feeling,'' he said. "I hope he does well, he's a top manager.''
Bellamy explained the chance to spend more time with his family was his primary motivation for joining Cardiff, and he does not expect to have it easy in the second tier.
"My wife and kids live here and I've spent the last six years travelling up and down the road,'' he said. "I left home at 15. I've always wanted to come back but I wanted to come back at the top of my game.
"I've come down to the Championship willing to put my neck on the line. It's probably the hardest football decision I'll have to make in my career. I grew up in this league and it's tough, and that gives me excitement as well. It's a great challenge for me.
"With the players we have I think we've got to be looking at promotion. I've been in the Premier League for 10 years now and probably now I'm at the height of my career.
"I feel in great condition but the next couple of weeks is going to be difficult - but that would have been the case wherever I'd gone.
"I don't mind dipping into the Championship for a year but hopefully I want to bring this club back to the Premier League. I've always watched Cardiff, whenever I've had an opportunity.''
Meanwhile, Celtic midfielder Scott Brown insists Craig Bellamy's decision to join Cardiff ahead of Celtic does not reflect badly on the Glasgow club or Scottish football.
He said: "I think it was more personal reasons for him not coming to Celtic. He just wants to go home and play in Wales.''
Brown does not believe concerns over the standard of the Scottish Premier League were a factor in Bellamy rejecting the opportunity to return north of the border.
"I don't think so,'' added the Scotland midfielder. "If you come to Parkhead and Ibrox and you see the fans and the games that we play, the games are decent to watch.
"We saw that with Hibs and Motherwell last year as well, in the game that finished 6-6. It's interesting to watch and, if folk don't want to come here, that's not our problem. With Craig, it was for personal reasons.''