The 32-year-old was released by Manchester United this summer following the expiry of his contract after making just 31 Premier League appearances over three injury-hit years.
Owen had been linked with moves former club Liverpool and their Merseyside rivals Everton before signing a 12-month deal at the Britannia Stadium on Tuesday.
"It's been a long summer, but I'm delighted and excited to be here," Owen told the club's official website.
|POTTERING UP THE TABLE
|16/1||Stoke City are 16/1 with PaddyPower to finish in the Premier League top six|
"I was taking my son back to school today and one of my daughters was going to school for the first time, so I said to them: 'It's like we're all going to school for the first day today!'
"That's certainly how it feels when you're meeting a lot of new people. It's only normal though, I've done it before and many players have done it before.
"It will take me a while to meet everyone and to bed in, but I'm certainly looking forward to it.
"I know how I feel, but in many ways, I'm unsure in terms of exactly what stage I'm at compared to the rest of the squad," he continued.
"They've had a long pre-season, so it might turn out when I do a few tests that I'm quite a bit behind or it might be that I'm further forward than I thought – I just don't know.
"I'll have some tests over the next few days and gauge where I am, and hopefully I'll be out there as soon as possible."
Owen went on to praise the Stoke fans, who regularly top surveys as the loudest supporters in England, adding that he cannot wait to play at the Britannia Stadium.
"Obviously, they don't get 70,000 here, but it's full every week and pound-for-pound is definitely one of the noisiest in the country," he enthused.
"I can only speak for when I’ve played here, but it's always been like that then. That's certainly part of the challenge for me here and something I'm looking forward to."
Following his first training session with Tony Pulis' side, the Chester-born forward later tweeted: "Baptism of fire in training today! Legs are stiff now but the club were happy with my level of fitness. May be ready sooner than expected!"
Owen could now make his debut when Manchester City visit the Britannia Stadium a week on Saturday.