Meeting the media at a Fenway Park news conference on Friday, Price, 30, said he was looking forward to all his new city has to offer.
"Boston was special to me, both good and bad, throughout my career," he said of his sometimes contentious relationship with the Red Sox and their fans.
"They have winning in their past," he said. "They've done it recently as well, and I really think we're going to do it in the future. That's what I want to be a part of."
The sides agreed to the terms of the deal Tuesday but had to wait for Price to pass his physical. The contract gives Price the right to opt out after three seasons with no deferred money. The yearly payout for Price is $30 million, $30 million, $30 million, $31 million, $32 million, $32 million and $32 million.
Price's deal was the richest ever for a pitcher until news broke Friday that Zack Greinke agreed to a six-year, $206 million deal with the Arizona Diamondbacks, according to ESPN and multiple media reports.
For all the money the Red Sox are paying Price, whom president of baseball operations Dave Dombrowski called "one of the best pitchers in baseball, a true No. 1," the Boston fans don't just want a pitcher who gives the team a chance to win every five days. And it's not enough for him to wind up among the Cy Young contenders in the offseason.
With a contract that large, the Red Sox are expecting playoff wins.
And that's the one thing Price hasn't delivered.
"I think I was just saving all my postseason wins for the Red Sox," said Price, who is 0-7 with a 5.27 career ERA in eight career postseason starts. "I know good things are going to happen to me in October. That just hasn't been the case thus far. I know I can throw the ball as well in October as I do in the regular season. That time is coming for me, and hopefully, it's in 2016."
Despite winning it all in 2013, their third World Series championship in 10 years, the Red Sox have finished last in the AL East in three of the past four seasons. Their rebuilding began in August, when they brought in Dombrowski as president of baseball operations, and this week he made his big move.
"Success in the postseason, sometimes it takes a while to come," Dombrowski said. "He's a big-game pitcher. [It] hasn't always happened in the postseason. I'm confident that he'll do that for us."
The deal with Price goes against the team's stated desire not to commit to long-term deals for pitchers who have already turned 30.
"There are exceptions to any rule, and certainly, this is one of the most exceptional pitchers," owner John Henry said. "He's putting up historical numbers, or at least bordering on that, at this stage of his career."
A five-time All-Star, Price won the AL Cy Young award in 2012 and has been runner-up twice in seven full major league seasons. He has a 3.09 ERA with 1,372 strikeouts and 104 wins. Last season, he went 18-5 with a 2.45 ERA with Detroit and Toronto, striking out 225.
Price's 1.95 ERA at Fenway Park is the best of any ballpark he has visited at least 10 times and the lowest among any player since Price made his debut in 2008.
"If I come anywhere close to meeting the expectations that I set for myself, I'm pretty positive that everybody will be happy with that," Price said.
Red Sox general manager Mike Hazen also praised Price's value as a teammate. While trying to decide whether to accept the ballclub's offer, Hazen said, Price asked about other players in the system all the way to Single-A Greenville.
When he was with Tampa Bay, Price complained about the way Ortiz admired a postseason home run before slowly trotting around the bases. Price hit Ortiz with a pitch the following season, the dugouts emptied and the two called each other names.
But Ortiz told the Boston Herald at his charity golf tournament in the Dominican Republic that his feud with Price was over. Price said on Friday he's looking forward to playing with "arguably the greatest DH to ever play the game."
"Big Papi and myself, we're both competitors. What he's done for this organization and the game of baseball is really special," said Price, who was given the No. 24 worn by Dwight Evans and Manny Ramirez. "I'm ready to be one of his really good friends."
Dombrowski, who worked with Price in Detroit, said he had no concern about him getting along with his new teammates.
"I had more than one person tell me, from a player perspective and from a staff perspective, that he was the best teammate they ever had," Dombrowski said.
The $31 million average annual value of the contract is second only to Greinke and matches that of Detroit Tigers slugger Miguel Cabrera, according to ESPN Stats & Information. Before Price's deal, the most the Red Sox had paid a pitcher, in annual value and overall cost, was the four-year, $82.5 million extension signed by Rick Porcello in April. Max Scherzer signed a seven-year, $210 million deal with the Nationals last winter (annual average value of $30 million), and Clayton Kershaw's contract extension with the Dodgers was for $215 million ($30.7 million).
"This is an exciting time," Henry said. "We're going to see one of the best pitchers in baseball every five days. It's going to be exciting for our fans, but it's going to be really exciting for our team."
The Associated Press contributed to this report.