The Rangers acquired the forward in a trade with the Tampa Bay Lightning on Saturday for a seventh-round pick in the 2022 NHL Draft.
Goodrow, who could have become an unrestricted free agent July 28, won the Stanley Cup the past two seasons with the Lightning.
"It was a pretty easy decision, I think," Goodrow said Friday. "The Rangers with such a storied franchise, the history here, the city, as soon as they showed a little bit of interest, I think that's all I needed. I was very excited at the possibility of joining this team right from the beginning, so I'm glad everything worked out and I couldn't be happier to be a Ranger."
The 28-year-old scored 20 points (six goals, 14 assists) in 55 regular-season games this season and six points (two goals, four assists) in 18 Stanley Cup Playoff games.
Goodrow has scored 93 points (32 goals, 61 assists) in 331 regular-season games with the Lightning and San Jose Sharks, and 14 points (five goals, nine assists) in 65 playoff games.
"When Tampa [Bay] traded for me (from San Jose, Feb. 24, 2020), they weren't looking for a guy who was going to come in and score 30 goals; they traded for the player who I was," Goodrow said. "I'm not signing here to be a player that I'm not. I'm not a guy who is going to score 30 goals or put up a ton of points. I'm going to be a guy who is going to be hard to play against, who is going to kill penalties, who is going to generate momentum for my team, who is going to stick up for his teammates, who is going to have leadership. I've learned what it takes to win at this level. I know the mindset you need to have to win, so I just look to share that as much as I can, to be a leader."
Rangers general manager Chris Drury said Monday that Goodrow's winning pedigree and ability to play anywhere in the lineup under new coach Gerard Gallant made him coveted.
"I think he's pretty versatile," Drury said. "I could see him do all different things for Gerard, whether that's more time in the middle, whether it's face-offs in the [defensive] zone. But we think he's got a pretty complete game. He's going be able to help a lot of different areas.
"He's a guy that we targeted and obviously went out and made a deal with Tampa [Bay] early to get a jump on it."
Drury said Goodrow fits into the Rangers' plans of developing a more straight-line approach because of his combination of skill, grit and north-south skating.
"Obviously the bigger the game, the better he seems to play," Drury said. "He's a winner. He's able to walk in our locker room and talk about what it's like to play in the playoffs and to win Cups. That's a unique thing and obviously a very special thing."
Goodrow arrived in New York on Thursday and said Rangers defenseman Jacob Trouba showed him around, answering questions he had. He said the first text message he got from a new teammate came from center Ryan Strome, who is from the Toronto area like Goodrow.
"I've played against 'Stromer' since I was probably 7 years old," Goodrow said. "We haven't played on the same team very often other than Under-17 Team Ontario, but he was one of the first texts I got. I called him just picking his brain on the city, the organization."
Goodrow's link to the Rangers began in 2012, when as a junior player he participated in their Prospect Development Camp and was supposed to return in September to go to the Traverse City Prospects Tournament, which never happened because of the NHL work stoppage.
He signed an amateur tryout agreement with Connecticut, the Rangers affiliate in the American Hockey League, in 2013 but never played.
Goodrow eventually signed with the Sharks as an undrafted free agent on March 6, 2014.
"I feel like I've always had ties to the Rangers," Goodrow said. "Going undrafted, finding a way to make it into the League and then winning two Cups, I do take a lot of pride in continuing to get better, but I always feel like I need to prove myself. I'll continue to do that with the Rangers."