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 to 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,” he told the club's official website.
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|16/1||Stoke are 16/1 to finish in the top six with Paddy Power|
“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.”
The former Liverpool and Real Madrid man, who has been plagued by injury in previous seasons, also admitted he is unsure how fit he is compared to the rest of his new team-mates after missing out on pre-season.
“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 City fans, who regularly top surveys as the loudest supporters in England, and admitted 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 added.
“I can only speak for when I’ve played here, but it’s always been like that. 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 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.