I was in a very similar position to the OP sometime back. I think it definitely depends on what your girlfriend is in to. Mine really doesn't care for action movies, so I decided not to show her a Klingon episode. I spent a looooooong time trying to figure out which one to show first. I think my top contenders were "Darmok," "The Inner Light," "The Measure of a Man," and "Data's Day."
In the end, though, I was unhappy with all of those for various reasons, so I decided to show her the first episode of TNG that I saw, which was "Tapestry." That seemed to work well enough. I think the second one I showed her was "The Measure of a Man," which also seemed to work well.
That said, I really have to agree with what R. Star
says below. After a few TNGs, I decided to show her a DS9, to give her a taste of my "other" favorite Star Trek series. I showed her "In The Hands Of The Prophets" first and she really
liked it. After "Duet," she was hooked. We are now working our way, slowly but surely, through the whole series (at her
request, no less!)
R. Star wrote:
I've converted far more people over to Star Trek using DS9. Unlike TNG you can show them the pilot without risk of prematurely scaring them away. That said, if you're determined to use TNG, there are a lot of good episodes to choose from. Really depends on what kind of person your girlfriend is, if she enjoys action sequences, character drama or comedy material. But a few episodes you can't go wrong with in my opinion..
Measure of a Man
Most of these are strong episodes that have little or no backstory that needs explaining beyond who everyone is.
Yup, these are all great episodes! I really like the suggestion of "Clues," in particular. It's a good "normal" TNG episode, as is "The Chase." But yeah, definitely agree with this list.
Third Nacelle wrote:
Data's Day is a very good introductory episode. It has an A-story that does a good job of showing many different aspects of the series and introduces the characters, and a fairly entertaining B-story.
First Contact (the episode), Who Watches the Watchers, The Chase, Lower Decks are all other good episodes you can watch without prior familiarity with the franchise.
I agree with "Data's Day," although I have some reservations because I think the business with the Romulans (and their relation to Vulcans) at the end might resonate less with a newbie. I'd be cautious with "Who Watches The Watchers," because of its religious content. YMMV. The only downside with "Lower Decks" is that the main characters are more-or-less one shots. But I agree that it gives a great look at the story-telling potential of the show.