In the first episode of her new Spotify podcast, Meghan, the Duchess of Sussex, explored feminism and the double standards faced by women, and she recounted her young son’s brush with danger while on an official tour in South Africa.

Her first guest on the podcast, which debuted on Tuesday and is called “Archetypes,” was the tennis superstar Serena Williams, who recently announced her retirement from the sport in a splashy Vogue cover story.

Meghan, 41, said she would use the series to dissect the labels and tropes that are often attached to women to hold them back. “Over the course of the next dozen episodes,” she said, “we’re going to live inside and rip apart the boxes women have been placed into for generations: boxes like ‘diva,’ ‘crazy,’ ‘the b-word,’ ‘slut.’”

Meghan, who was the subject of intense and frequently racist scrutiny by the British tabloid press, said, “I know a thing or two about these labels myself.”

“My hope is that my own lived experience will help other women open up, to reveal the layers that thrive within all of us and the truth that none of us are alone in this world,” she continued. “That the future is something we get to write together.”

In an hourlong episode, Meghan and Ms. Williams discussed the label “ambitious,” and how dehumanizing it can feel when men are labeled driven in their careers and lives, whereas ambitious women are often labeled selfish and called social climbers with hidden agendas.

“I don’t ever remember personally feeling the negative connotation behind the word ambitious, until I started dating my now-husband,” Meghan said, referring to Prince Harry. “And apparently ambition is a terrible, terrible thing — for a woman, that is, according to some.”

According to Meghan’s other guest on the episode, Laura Kray, a professor at the University of California, Berkeley, the way ambitious women are seen as power-hungry and manipulative, while ambitious men are seen as being role models, “boils down to antipathy toward women.”

“Ambitious women who have children and then come back to work and attempt to perform at the highest level are violating people’s expectations of what the most important thing for women to be is,” said Professor Kray, an expert on barriers faced by women in the workplace.

Last year, Meghan and Harry gave an explosive interview to Oprah Winfrey in which Meghan accused the royal family of being callous and racist after the couple’s marriage in 2018. The pair moved to Montecito, Calif., where they currently live with their two young children.

Meghan recalled a scary incident that happened while she and Harry were on a tour of South Africa with their son, Archie, who was 4½ months old at the time. She said that as soon as they arrived in Nyanga, a township in Cape Town, they had to leave the baby with a nanny at a housing unit where they were staying while they went to an “official engagement” where Meghan was to give a speech to women and girls.

When she finished and returned to the car, she was told that there had been a fire in the nursery, in which there were no smoke detectors. Thankfully, just before, Archie’s nanny had taken him with her to grab a snack downstairs, but Meghan was extremely shaken up.

“The heater in the nursery caught on fire,” she said. “There was no smoke detector. Someone happened to just smell smoke down the hallway, went in, fire-extinguished. He was supposed to be sleeping in there.”

“And of course as a mother you go, ‘Oh my god,’” she added.

Having never heard the story before their chat, Ms. Williams seemed shocked.

“I couldn’t have done that,” Ms. Williams said about the fact that Meghan had to continue on with her official duties. “I would have said nuh-uh.”

“These human moments behind the scenes, the ones under the surface, they’re everything,” Meghan said. “Because when we don’t swim in the shallow end, and instead choose to dive into the deep end, that’s when we gain a more nuanced understanding of each other.”

Ms. Williams, 40, also shared a shocking experience she had the night before a match at the French Open in 2018, when her daughter, Olympia, fell out of her high chair and broke her wrist. After a trip to the hospital, during which Olympia’s wrist was put in a cast, Ms. Williams spent most of the night awake, rocking her to sleep, she recalled.

“She was on my watch, and I was just basically devastated,” Ms. Williams said. “Like, I literally couldn’t think. I felt so guilty.”

“I somehow managed to win,” she added, “but I was so emotionally spent and so emotionally drained that it was crazy.”

That was the same match at which Ms. Williams, a 23-time Grand Slam winner, wore a black catsuit for medical reasons, leading to both praise and outrage. (The French Tennis Federation later banned such outfits.)

When it came to her retirement, Ms. Williams said she liked to think of it as more of an evolution, adding that she would always have some involvement with the sport as she pursued other business ventures and opportunities.

Ms. Williams also said that she really wanted to expand her family.

“I’ve been putting it off for so long, and as a woman, there’s only so, so long you can put that off,” she said, adding, “I’ve been fortunate enough to play tennis really well, but I think my best is being a mom.”

Prince Harry also made an appearance on his wife’s first podcast episode, just before her interview began. Meghan and Ms. Williams can be heard saying hello to Harry in British accents.

“I like what you’ve done with your hair, that’s a great vibe,” he said to Ms. Williams, presumably via video call.

In December 2020, months after the two gave up their royal titles, Spotify announced that the company had agreed to a multiyear partnership with Archewell Audio, the audio-first production company the couple created to produce podcasts. The couple also signed a multiyear deal with Netflix and announced their first project, “Heart of Invictus,” in April 2021.