"Today is my first day at work, and there's a lot of work to be done," Shanahan said.
"I think that the opportunity to win is special in any place. Whoever wins the Stanley Cup this year, it'll be a special occasion. I recognize, and everybody in hockey recognizes, what potentially it could be like in Toronto.
"This is the time for me to start learning about the organization from top to bottom. It's a time for me to listen, to learn and get to work."
The Leafs have several free agents to focus on this off-season, though many believe pending unrestricted free agent right wing Nikolai Kulemin will not be re-signed.
Forwards Dave Bolland and Mason Raymond both expressed their desires to stay in Toronto, in speaking to the media prior to Shanahan's press conference, though it's unclear if either will be re-signed.
Bolland is believed to be seeking a deal worth around $5 million per season over a six-to-seven year term, a length the Leafs are not believed to be interested in.
Shanahan, Nonis and the Leafs' management staff will be dissecting the current roster over the next couple of weeks and determining what direction to take with certain players.
"I do like a lot about this group. From an outsider looking in, there are some really good pieces that a lot of teams would covet," Shanahan said. "There are some great aspects to this team.
"(But) if you have an opportunity to make improvements, you make them."
In addition to free agency, the Leafs are expected to be active on the trade front, with the likes of goalie James Reimer and center Nazem Kadri potentially on the move.
"This team is not where it needs to be," Nonis said. "There's a lot of things that we have to change and a lot of places we have to build upon and to go if we're going to have a chance to win, and itís not going to be easy."
Nonis and Shanahan must also decide the fate of head coach Randy Carlyle, who has been rumored to be on the hot seat for some time. Both men will meet with Carlyle in the near future and discuss next steps, whether he's coming back next season or not.
"I don't think it's the appropriate time to give an assessment of players or coaches," Nonis said. "I think it's important now we step back and have a thorough review of the entire organization, coaches included. We'll do that over the next couple of weeks.
"I think Randy Carlyle is a good coach. That's as far as I'll go."
TFP Editor-in-Chief David Pagnotta contributed to this report.