The goalkeeper has moved to the top-flight side on a four-and-a-half-year deal for an undisclosed fee, understood to be worth just under £5million.
Butland turned down the chance to talk to Chelsea because he wants first-team football and is delighted with the transfer.
He told Sky Sports: "I feel very good, glad to get it done and it's the next step of my career. I'm really looking forward to it.
"I'm glad something's happened now. I feel very lucky to come to a great club and I'm looking forward to it.
"Stoke just ticked all the boxes for me. I've come to a great club, a very solid club which is doing very well in the Premier League with great support as well."
The young England international, who has been called up for next week's friendly with Brazil, paid tribute to Stoke's staff.
He added: "I can see its very professional and a very honest environment which I'm looking forward to getting me teeth stuck into and working with them."