The 21-year-old midfielder played 45 minutes in each of their friendly victories over FC Gravenzande and Excelsior following his £5m transfer last week.
"I watched Swansea play a lot in the Premier League. The way they play football excites me and I can't wait to be part of it," he said.
"When I spoke to the Swansea chairman (Huw Jenkins), he said they had been tracking me from my Charlton days. I also heard of a link when Joe Allen was going to Liverpool, but that never happened.
"It raised its head again over the summer and I was following it in the papers every morning while on holiday until it happened for real."
Shelvey has received a warm welcome from the rest of the Swansea squad, and he added: "It's always a nervous time joining a new club. It was the same when I went from Charlton to Liverpool, then to Blackpool on loan and now Swansea.
"You don't know what to expect meeting the players for the first time, even down to the staff. But I have learnt that you can't be negative. You have got to be positive and confident in order to show your character and ability.
"In fairness to the boys here, they welcomed me in straight away. I had heard that Swansea was a friendly club. I spoke to Joe Allen just before I signed and he had nothing but good things to say about the club and the fans."
Shelvey is also looking forward to the challenge of playing in the Europa League with the Swans this season as Capital One Cup champions.
He said: "It's exciting for the fans and I'm sure the atmosphere will be buzzing at the Liberty for the night games. It will also give the players experience of going to different countries and facing different styles of play."