The 32-year-old former England striker quit the Potters - and the chance to remain in the Premier League - for a move to the Scottish champions after penning a two-year deal, with a further one-year option, on Friday.
It proved to be a whirlwind 24 hours for the player, who was given a tour of Ibrox on Thursday morning before thrashing out the deal which sees him become the club's first purchase in two years.
Beattie said: "I've never played Champions League football before so I think that is a step up and a progression.
"The size of the club is huge and I think it's a massively positive step for James Beattie and for Rangers.
"I've had a lot of text messages from friends that I didn't realise I had saying 'you've just signed for the biggest club in the world' and stuff like that.
"I'll probably get to know how big the club is as I play.
"There was interest from other clubs but when you come up here and you speak to Walter Smith and have a look around, the decision was easy."
Beattie featured sparingly for Stoke in the second half of last season following an altercation with manager Tony Pulis and he admits he jumped at the chance to move to Rangers.
He said: "I'm delighted to sign for a club like Glasgow Rangers and it was quite a quick deal as far as I'm concerned.
"I came to Ibrox on Thursday morning and was met by the gaffer and (chief executive) Martin Bain.
"I was shown around the stadium, which was magnificent, it was the first time I'd been there. You can see it is a club steeped in history.
"I then walked out on to the pitch, which was magnificent. The walk out, even when the stadium was empty, was something so I'm looking forward to walking out when it's full.
"I had quite a long chat with the gaffer and he spelled out what he wanted me to do and I told him what I wanted to come here and do.
"The guys then did the numbers, which was quite quick, and then it was down to the two clubs to settle on a fee and I'm glad to say that's all finished now and I'm delighted I've signed."