Saldana does a full-voiced rendition of the show's opening fanfare. "I would look over and all these three women were glued to the television, watching the series in Spanish. It's so funny. It was the first series that I would experience vicariously through their eyes."

Twenty years after her first exposure to Star Trek, the US-born, half-Dominican, half-Puerto Rican actress got her big break in JJ Abrams's 2009 blockbuster reboot of the series, playing Starfleet communications officer Uhura.

Saldana, now 37, legs folded beneath her on a sofa at the St Regis hotel in Atlanta, puts it all down to fate.

"This was something that the universe and I were working in conjunction to make happen," she says. "Every now and then I have moments in my life when I look up and say, 'How did you know I wanted this so bad'?"

Though she starred in the teen ballet drama Center Stage and the first Pirates of the Caribbean movie, for the last few years, Saldana has spent most of her screen time exploring the outer reaches of galaxy. As well as Star Trek, 2009 saw the release of James Cameron's Avatar, in which Saldana played Sam Worthington's blue-skinned humanoid alien love interest Neytiri.

"It was the most amazing, challenging, scariest experience I've ever had approaching a character and trying to conceive a character," Saldana says. "I really wanted to explore it and do, not my best, but do it perfectly."

Saldana was immersed in the project for the better part of 18 months. "I didn't go to Juilliard or Yale to study acting. This was the best form of schooling I ever had."

More recently, Saldana traded the gentle baby blues of Avatar's Na'vi for praying mantis-green, playing the butt-kicking alien assassin Gamora in Marvel's Guardians of the Galaxy.

It's not easy, it turns out, being green: as Gamora, Saldana is regularly picked up at 2.30am, 3am to face endless hours in the make-up chair.

Today's interview-day version Saldana, though, is thoroughly human, sporting a red Chinese tangzhuang and clearly relieved to be wearing her own face. "Guardians makes me miss every other character I've done in every other movie," she says. "It makes Uhura seem so low-maintenance."

Saldana is in Atlanta straddling a couple of dimensions at once: promoting the third Star Trek instalment, Star Trek Beyond, opening in July, while shooting the second Guardians of the Galaxy movie.

Turning her acting career into a game of interstellar hopscotch was never a conscious decision. "One thing simply kept leading to another," she says.

She does admit, however, to being an effusive admirer of the ambitious world-building film projects of the big-name directors she's worked with: Abrams, Cameron, Spielberg. She thrills at the feeling of "stepping into someone's creation" and has a natural preference – it's her "guilty pleasure", she says – for stories set in the distant future over those set in the past.

Put it down to an imagination that operates at warp speed, she says. "On a personal level, it's what I like. It's the literature I like to read. It's not that I'm ahead of my time, but … The future moves at a pace to my liking. It moves me a great deal. It's so boring to read things that are gone."

Saldana traces her love of the fantastical to her childhood in New York City. (Saldana's family moved to the Dominican Republic when she was 10, after her father died in a car accident.) She cherishes the memory of the bedtime stories her mother would tell her and her two sisters. "She had a beautiful way of telling a very simple story." But Saldana was also exposed to an unusual amount of mind-expanding science-fiction.

"New York wasn't as safe as it is today," Saldana says. "So we spent a lot of time in our little apartment. We watched everything. Blade Runner, Aliens, Stanley Kubrick's 2001: A Space Odyssey. It made that little space feel so huge for us. It wasn't like, because you're girls, let's watch this boy band." After a brief pause, Saldana adds: "We would also watch The Last Unicorn."

Spanish-dubbed reruns of Star Trek didn't interest 10-year-old Saldana as much as they interested her older family members. By the '80s its portrayal of futuristic technology felt, to her, dated and goofy. It was Saldana's mother who informed her she was missing the point.

"My mother told me: 'You don't understand'," says Saldana. "'Don't look at the show. Observe the message. Look at all those beautiful colourful people working together in the future. It is possible'."

For all its shiny depictions of the 24th century, perhaps the most evolved feature of Star Trek was its utopian vision of cross-race, cross-gender, even cross-species harmony. The USS Enterprise boasted a Japanese-American lieutenant, a female African-American communications officer and, in Spock, a Vulcan-human first officer. When Nichelle Nichols, the African-American actress who played the part of Uhura in the original series, met Martin Luther King Jr, he declared himself "the biggest Trekkie on the planet". "You are changing the minds of people across the world," he told her, "because, for the first time, through you, we see ourselves and what we can be."

Saldana has spoken frequently about dealing with the ageism and sexism, but particularly, the racism of Hollywood. With her sisters, she launched Cinestar, a production company devoted to producing Hispanic content. Hollywood could learn a thing or two from the standards of inclusivity on the USS Enterprise, she says.

"We have to let art accurately and bluntly imitate life. And also inspire life. A show like Star Trek, with its concept and its mission of peace, is super-inspiring."

That said, for all the starry-eyed optimism of the original Star Trek, '60s America wasn't yet ready to see a television romance between a white man and a black woman. A romantic relationship between Uhura and Spock was hinted at but never made overt.

In the new Star Trek movies, Abrams led the way in realising that most sub-textual of subplots. "It's super important to humanise leading characters," Saldana says. "They're working together for years, sometimes on five-year missions. You mean to tell me no one's going to fall in love?"

In recent months, Saldana has been hauled back down to Earth in brutal fashion. For her performance as the African American singer Nina Simone for the 2016 biopic Nina, the lighter-skinned Saldana had her skin darkened – the term "blacker-face" featured in some of the more disdainful write-ups – and wore prosthetics.

Saldana was very much in the firing line in the subsequent debate, accused of being yet another lighter-skinned actress appropriating a role that rightly belonged to a darker-skinned performer. Simone's daughter herself weighed in on the issue: "Appearance-wise this is not the best choice," she said.

Before my time with Saldana, two minders separately advise steering clear of the topic, warning that it's likely to bring the interview to a premature halt. For her part, Saldana brushes aside those concerns, and responds carefully, but with conviction.

"It was a big learning experience for me," she says. "I'm happy that we are talking about this matter as a human society, as an American society, as an artistic society. But …" she pauses. "Being violent, and hateful, and judgmental about another person … Being hateful and negative will not solve the situation."

Nothing will keep Saldana from owning her own cultural identity. It's a responsibility she feels more deeply, now that she's a mother with toddler twins.

"I love my culture," she says. "I love being a first generation Latina, I love being a Latino woman, just as much as I love being an American woman. You have to embrace this world that you belong to, but you have to learn all the beautiful traditional things that your family has brought with them and are still holding on to them. You're the one that will give them that longevity."

Now, Saldana's career trajectory is pointed upwards again, at the stratosphere. The next Guardians of the Galaxy will arrive in 2017. James Cameron, meanwhile, has promised no fewer than four Avatar sequels.

Apart from anything else, it sounds like an opportunity for Saldana to dust off some of the words and phrases she picked up on Avatar, speaking the made-up language of the Na'vi. One that she has retained is "Oei ngati kame". "It means 'I see you'," Saldana explains, but also "I comprehend you", a serenely positive sentiment of peace and tolerance.

The transcendent wisdom of an imagined future obviously jibes well with Saldana's own philosophy. "I can't wait to go back," she says.