Prince Harry 'Can't Wait' to Take His Kids to Invictus — and Reveals Lili 'Just Took Her First Step!'

Prince Harry is one "proud papa."

The Duke of Sussex may be cheering on athletes across the ocean at the Invictus Games, but his children with Meghan Markle — Archie Harrison, who turns 3 on May 6, and Lilibet Diana, 10 months — are never far from his mind...especially after his daughter's latest milestone!

"Her current priorities are trying to keep up with her brother; she took her first step just a few days ago!" Prince Harry, 37, tells PEOPLE in this week's exclusive cover story. "Proud papa, here."

When it comes to bringing Archie and Lili to the Invictus Games, an adaptive sports competition for veterans and service personnel, Harry says, "We can't wait!" — and he's already introduced the competition to his son.

"I showed Archie a video of wheelchair basketball and rugby from the Invictus Games in Sydney, and he absolutely loved it," Harry says.

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"I showed him how some were missing legs and explained that some had invisible injuries, too," he continued. "Not because he asked, but because I wanted to tell him. Kids understand so much, and to see it through his eyes was amazing because it's so unfiltered and honest."

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Prince Harry's personal life has changed greatly since launching the Invictus Games in 2014, most notably becoming a husband and father.

"Being a dad certainly adds another emotional layer to it," says Harry. "When I was in the Army, I promised myself I would be out before having a wife and kids, because I couldn't imagine the heartache of being apart for so long during deployment, the risk of possibly getting injured, and the reality that my family's lives could be changed forever if that happened."

Chris Jackson/Getty Prince Harry

He adds, "Every member of the Invictus community has experienced varying degrees of these things. I have tremendous respect for what they and their families sacrifice in the name of service."

Chris Jackson/Getty Images for the Invictus Games Foundation Prince Harry

Invictus has also played a key role in Harry and Meghan's relationship — the pair made their first public appearance together at the Toronto Games in 2017.

"There is nowhere you can feel more embraced and supported than with the Invictus family," he says. "The Toronto Games were our first time out and about publicly in an official way. We were dating at the time, so it was a lot to take in, but fortunately, we were with the perfect community for that.

"Now, five years later, here we are in The Hague at the fifth Invictus Games, as parents of two, and living in the U.S. I had always wanted to share these incredible moments with someone special, and to have Meg by my side means everything."

Patrick van Katwijk/Getty Images Meghan Markle and Prince Harry

At the opening ceremony on Saturday, Prince Harry told the crowd about Archie's potential career aspirations.

"When I talk to my son Archie about what he wants to be when he grows up, some days it's an astronaut, other days it's a pilot — a helicopter pilot obviously — or Kwazii from Octonauts," Harry said with a laugh, adding, "if you're laughing, then you've seen that."

"But what I remind him is that no matter what you want to be when you grow up, it's your character that matters most, and nothing would make his mum and me prouder than to see him have the character of what we see before us today," he continued.

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And when young reporters Sophia and Jay of Kindercorrespondent (which translates from Dutch to Children's Correspondent) had the chance to interview Prince Harry, they asked him about his wish for his two little ones.

"To grow up in a better world," Prince Harry replied. "To grow up in a fairer world, a safer world, a more equal world. It's not going to be easy, but I will never, ever, ever rest until I as a parent have at least tried to make the world a better place for them. Because it is our responsibility that the world is the way that it is now. I don't think we should be bringing children into the world unless we are going to make that commitment to make it better for them."

He added, "We cannot steal your future."