Obviously, it depends on what your girlfriend likes, and what you're comfortable running.
From what you've said, I'm not sure what level of gaming experience she has firsthand. If none, then I'd make sure that I put her in something kind of generic, so that she didn't get the weird of experience of being in a universe she recognizes but not getting to do the things that the characters in that universe can do. (For example, if you put her in a Star Wars universe, she might be annoyed at not getting to do all the cool Jedi tricks right away.)
I'll go a bit different and suggest that you use Mutants & Masterminds -- and that you use it to create a kind of pulp-space-opera world that is familiar to her as similar to Star Wars or Buck Rogers or whatever, but different enough that she isn't tripping over the rules of doing what she saw on the show.
The bad news on M&M is that character creation is a royal pain the first few times, which is why YOU would create her character with her, letting her describe the character and then coming up with the rules for that character -- what powers, feats, and skills the character would have.
In terms of people versus ships, you could essentially have two games that run in parallel -- ship-scape and people-scale. If you said that Kirk was PL10 and had knowledge, fighting ability, the ability to inspire his crew, and the techno-gadgets that he always has (communicator, phaser, etc), while McCoy has healing and his medical tricorder and resistance to mental domination, and Spock has telepathy and all that, that would be the people-scale. And then, on the ship-scale, the Enterprise has Flight (for impulse power), Space Flight (for warp speed), Energy Blast (phasers), Force Field, and a few other goodies, and you could put that at PL10 as well -- with the understanding that ship-scape PL10 is always going to be more powerful than people-scale PL10.
But that's rules-stuff. That's your end. Once the world is created, once the characters and ships and whatever are in place, M&M is really really easy to play, and it allows for much more creative uses of powers than, say, D&D spells. I think it would be fast enough and easy enough that, provided YOU made the character, it'd be a great way to introduce someone to roleplaying.
This is provided that she's naturally a rules-light person (which is what I got from "I'd get bored doing that," although that could relate more to a game she saw that involved killing things and taking their stuff). If she's a CPA or computer programmer, she might love a more rule-based system.
