The 34-year-old, who won four Premier League titles, the EFL Cup five times and the FA Cup twice during his time with City, agreed terms on a two-year deal with the Basque club earlier this month.
Sociedad finished sixth in LaLiga in the 2019-20 campaign, though they won just two of their 11 fixtures following the restart.
"I'm really excited to get started here and meet my new team-mates and the club staff," he told a media conference.
"The team is playing some great football and I hope to add to that here and have some great years at the club.
"I have high expectations in terms of the football we'll be playing and I think we can compete for silverware.
"I'll be an important part of the team and in this case, because of my experience, I'll help the younger players as well.
"I've been lucky enough to win a lot of things in my career, I'm very competitive and I'm going to try and do the same here because that's why I've been brought in."
"Of course, after so many years playing in Europe, you want to continue doing that," he explained.
"I also came because I like the style of football the team plays. I also came for the relaxed way of life here which I was already familiar with having lived in the Basque Country years ago."