"At every level, [Miller has] had the opportunity to play in the playoffs, especially the last two or three seasons, getting that much playoff experience, playing in those situations, the intensity, just having that understanding of what it takes to get to the playoffs and to have success in the playoffs."
Miller, who was acquired from the Boston Bruins in the 2017 NHL Expansion Draft, had NHL career highs in goals (10), assists (31) and points (41) in 82 games with the Golden Knights in 2017-18, helping them reach the Stanley Cup Final.
He said he enjoyed his time in Vegas, but is eager to start the next chapter of his career wth the Sabres.
"It's part of the business," Miller said of being traded. "It happens. I don't think I'm looking at it as, I know I won't be looking over my shoulder now at what Vegas is doing. I think you're focused now on the opportunity that you have moving forward. It's definitely a good one here. You're just excited for that and go on from there.
"I don't have any connections, but it was always a team that was talked about in my hometown, stuff like that, because they were so close. I'm from Sault Ste. Marie [Ontario]. It's not very far. The closest teams are [the] Toronto [Maple Leafs], Detroit [Red Wings] and right there, Buffalo. It was always a team that you'd catch a game or whatever when you're a kid growing up. So I'm excited. It'll be fun to put on my jersey."
Selected by the Los Angeles Kings in the fifth round (No. 151) in the 2012 NHL Draft, Miller has 99 points (22 goals, 77 assists) in 250 regular-season games and 11 points (four goals, seven assists) in 30 playoff games.
The second-round pick Buffalo traded was acquired from the St. Louis Blues.
Golden Knights general manager George McPhee said he does not expect a big free-agency splash.
"I don't anticipate a whole lot more," McPhee said. "We're in a good place right now. We'll see what's out there. As you all know, we're not big-game hunting, we're not going to be aggressive in free agency. Ideally, you'd like to be in a place in free agency where you just go fishing. We can sit back and see what's available and see if there's anything that makes sense for our team. If there isn't, we're fine."
Vegas has five defensemen under contract for next season, including Nate Schmidt, Shea Theodore, Brayden McNabb, Nick Holden and Jon Merrill.
"We have some very talented young defensemen," McPhee said. "Whether it's one guy playing here all of next year or five guys getting 15 games each, we'll see how it goes. But it's kind of nice to know we've got a spot and people are going to audition for it and see who grabs the brass ring."
McPhee said the Golden Knights are hopeful of signing veteran defenseman Deryk Engelland; the 37-year-old can become an unrestricted free agent on July 1.
"We're in a pretty good place there," McPhee said. "We're just trying to work out some of the bonus stuff, but we're in a good place with him. We're pretty much clear on who will come back."
NHL.com correspondents Heather Engel and Danny Webster contributed to this report