Graham will go into the Premier League as Swansea City's record signing after Brendan Rodgers parted with £3.5m to bring the striker to the Liberty Stadium from Watford.
And it marks a return to the top flight for Graham, who was just 19 when he appeared there with Middlesbrough.
“It was tough for me at the start with Middlesbrough, I was young and we had Jimmy-Floyd Hasselbaink and Mark Viduka at the club
“I was learning a lot and enjoying myself but not getting many chances in the first team, just a couple of minutes here and there.
“I made a big decision to leave and my plan was always to work my way back to the Premier League because that’s where every footballer wants to play.
“I’m much better now than I was at 19,” said Graham. “I’m more confident and, importantly, I’ve got a lot of goals behind me.
“I always had confidence in my ability and that I could make it in the top flight. Of course you never know how things are going to work out and I did wonder sometimes if I would get back to the Premier League.
“Like every striker I go through spells when I’m on fire and others when I don’t score for a couple of games, the difference now is I don’t worry about it when I am not getting goals for a couple of games.
“This is a big move for me but I am at the age when nothing really fazes me any more.
“Carlisle to Watford was a big move and I coped with it so I am just looking forward to getting started with Swansea in the Premier League.
“Looking from the outside, Swansea are successful as a club because there are no big egos and the focus is more about the team. That’s perfect for me.
“It’s very similar to what we had at Watford under Malky Mackay and I loved being part of that.
“Marcos Senna is a top player and it’s great to read people like him are interested in coming to Swansea.
“I’m not surprised because when I spoke to the manager I was so impressed with what he had to offer. He said everything I wanted to hear and after that I knew straight away where I wanted to go.
“He is very driven and so am I so hopefully we will work very well together.
“To be honest I’m looking forward to playing everyone,” he said. “They are all big games.
“I’ve played against Swansea four times in the last couple of years so I know what we are all about.
“Everyone saw last season how many goals Scott (Sinclair) got from out wide and Nathan (Dyer) was player of the year, so I’m really looking forward to the service they can give you up front.
“Since I signed the fans have been sending me plenty of messages of support on Twitter, and I’ve had a bit of abuse from Cardiff fans as well so I’m feeling like a proper Swansea player already,”