The Wales international, 26, joined the Potters for £13m in July after four seasons at Anfield.
"Everything's been positive," Allen told BBC Radio Stoke.
"Stoke and Liverpool came to agreement very quickly and it was all very smooth - that's been great because I've had a couple of weeks with the team ahead of the new season."
Allen arrived at the Bet365 Stadium on the same day as Egypt winger Ramadan Sobhi, with the pair becoming boss Mark Hughes' first signings of the summer.
And Allen says the calibre of players already at his new club meant he was always confident he was coming to a serious team.
"You look at the quality that's already here, like Glenn Whelan and to get Glen Johnson was a real coup," he added.
"With the ones they've recruited in the last few seasons, like Shaqiri, Bojan, Imbula, Arnautovic and Jack Butland pushing for the England spot in goal, it's a squad that's got quality throughout.
"I'm hoping to add to that and I certainly don't see leaving Liverpool and joining Stoke is a big drop - I wasn't expecting it to be."
After three successive ninth-placed finishes, Stoke have become regulars in the Premier League's top 10.
And Allen says that consistency under Hughes was one major pull.
"There were so many factors," Allen said. "The stability that they've had over recent years, the manager and his staff have improved the team and the players I'd get to play with meant it was a no brainer for me.
"I trust myself to give 100% effort and that's what my focus is on - I can't wait."
Midfielder Joe Allen says he does not see his move from Liverpool to Stoke City as a "big drop" in his career.