Kenan Thompson is one of a group of celebrity fathers who stars in the new documentary film Dads, available now on AppleTV+
Kenyan Thompson and daughter
Kenan Thompson and daughters

With over half a decade of parenting under his belt, Kenan Thompson has some advice for dads-to-be: Get ready for your life to change in an instant.

"I think the biggest wake up call is how immediate it is," the Saturday Night Live star told PEOPLE of fatherhood in an exclusive chat surrounding his new documentary Dads, available now on AppleTV+ just in time for Father's Day.

"One day you're not a dad and then the next moment you are. You're just a dad from that point on," adds Thompson, 42, whose older child Georgia Marie turned 6 on Saturday. (He and wife Christina Evangeline are also parents to daughter Gianna Michelle, 2 next month.)

"And definitely, my coolness just got sucked right out of me," he jokes. "I didn't know... I didn't even feel it, but I just immediately started wearing black socks and shorts and just being really corny at stores and over-asking questions about items that are super obvious and in my face."

Thompson says that having a second child "kicked [things] into overdrive," continuing, "I'm so thankful they weren't in diapers at the same time" — something he calls one of "the hardest things to do on the planet."

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Kenan Thompson Celebrates His Daughter's Birthday At Walt Disney World
Kenan Thompson and daughter Georgia
| Credit: Chloe Rice/Getty

Thompson says having a child for the first time is "such a learning experience" that is challenging despite having access to "books and apps and stuff to guide you through the ups and downs" and "all the milestones that you're supposed to be looking for."

"The constant worry though," notes the All That star, "is learning how to calm down and to see how resilient kids are — learning what the real warning signs are, as opposed to just being worried about every little slip and fall."

Luckily for Thompson, he has Georgia to help guide her younger sibling. "It helps to have an older sister to run around and teach her those things 24/7 that we can't necessarily communicate the same kind of way," he says. "[Gianna will] just pick things up watching her sister, basically — how to walk up and down the stairs."

"And she's getting so long. It's like you really see it when they stretch out for naps and stuff, just flat on their back," the actor adds of Gianna. "I haven't had that experience since Georgia, but they're four years apart, so it's been a while. But to see her coming along like that ... because kids, in their first year, are exhausting. They really are. They need everything from you."

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Kenan Thompson
| Credit: Dimitrios Kambouris/Getty

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Discussing his involvement in the new Bryce Dallas Howard-directed documentary — which also sees appearances from celebrity fathers like Patton Oswalt, Will Smith, Jimmy Fallon and Neil Patrick Harris — Thompson says becoming a dad himself "definitely stamped my existence outside of myself."

"It just all happened at kind of the same time. We were married for three years before we had a baby, but it was always in our mind. So it was always that plan," he tells PEOPLE. "The whole family-man thing of it all just kicked right in, and it's just been an amazing ride."

"I've definitely grown and matured in ways that I never imagined," Thompson adds. "And then there were ways that I did imagine that come through, like dropping my kids off at school and what that experience was like, and having things come full circle like that — like watching a big girl do her homework. It's just been amazing."

Dads is playing now on AppleTV+.

For more from Kenan Thompson, pick up this week's issue of PEOPLE, on newsstands now.