Johnson, 25, had been a bit-part player for the champions last term and was mainly restricted to substitute appearances, meaning he missed out on England's Euro 2012 squad.
However, he started in this month's 2-1 friendly win over Italy to pick up his 11th cap.
"Adding quality players to the squad has been our main aim this summer and Adam certainly fits that bill," Sunderland's ambitious manager Martin O'Neill said on the club's website.
"He has terrific ability, great delivery and I'm sure he is a player who will excite our fans. I couldn't be more delighted to have him at the football club."
Tricky wide-man Johnson, who started his career at neighbours Middlesbrough before signing for City two years ago, has joined for an undisclosed fee.
The 25-year-old Fletcher, who handed in a transfer request at Wolves this month after they were relegated from the Premier League last season, has also moved for an undisclosed fee.
"It has been no secret that we needed to strengthen our attacking options and Steven has a proven track record of scoring goals," O'Neill said.
"He is young and constantly improving and he will be a very welcome addition to the squad. I'm delighted that we've been able to bring him to Sunderland."
Fletcher scored 12 goals in each of the past two seasons for Wolves.
He was the second player to leave Molineux on Friday, with winger Matt Jarvis also returning to the top flight with West Ham United.
Sunderland's investment follows a strong 13th place finish in the Premier League last term after former Aston Villa and Celtic boss O'Neill was appointed last December.
