The 24-year-old will compete in the Renault Series 3.5 next season, bringing his ambition of a drive in Formula 1 a step closer.
Stoneman finished runner-up in last year's GP3 Series, and has worked as a test driver for Williams in F1.
"I don't dwell on my illness, it's behind me. This is the year to turn my career round," he told BBC South Today.
"It's going to be good. Hopefully I can have another good season. I had a fantastic season in GP3 last season and I just want to continue that again this year.
"Red Bull have got a fantastic reputation and it's been my dream ever since I was a little kid to be in with them and now it's come true.
"Carlos Sainz raced in this championship for Red Bull last year and won it. And now he's in F1 - this is my time. It's a turning point.
"Obviously to be a Red Bull driver and to be on their simulator and so on, they're looking to progress me on but there's no guarantee I'll be in F1."
Southampton-based Stoneman met with Helmut Marko, the head of Red Bull's driver development programme, to confirm his move to a junior programme with a remarkable record of producing Formula 1 drivers, with four-time world champion Sebastian Vettel, Daniel Ricciardo and Sainz among its graduates.
"He's a nice guy. The meeting was so relaxed - basically he likes speed and wants a driver who's going to win," said Stoneman.
"I've got the determination to go out there and do a job for myself and the team. He's looking at the overall package and the potential to progress through the categories."