Not sure this is any help. But a while back I thought my young nephews should learn about Star Trek. They had watched very little of any sort of TV, so they were up for anything, but I wasn't sure where to start. I thought classic Trek would be too old or dated for them, but they could appreciate it later, so I decided they should start with TNG because it may be more accessible for them.
But where should they start in TNG? Yes, we could start with Encounter at Farpoint, the first episode... But that is a pretty cheesy, crappy, strange episode(s). Right? Though it does introduce Q, who is a recurring 'integral' 'villain'. As a completist, you really would have to start here. (Or with TOS.)
Or I could just jump right in and show basically the best episode (or at least best cliffhanger) of Star Trek ever, The Best of Both Worlds I & II. But would it be as good an episode if they don't know any of the characters at all, and never previously encountered the Borg. In fact, so much would be lost by showing that episode first, I decided we couldn't.
So, stay with me here-- I decided to start them with Coming of Age. Now, yes, this is a Wesley episode. And it's near the end of the first season. But Wesley is not at all insufferable in this episode-- he's actually interacting in a more or less reasonable way with his fellow candidates, and my nephews are of the age (12ish) where they could actually maybe see themselves in his place, going through those tests. And while I skipped over a lot of (bad) first season episodes to get to that one, I thought I shouldn't skip the entire first season. That wouldn't be right! So this is a compromise. But most of all, the episode actually does a fairly good job of introducing the characters, through Remmick's interviews/interrogations, and we see Picard have to make a big choice; to become Commandant of Star Fleet Academy, or remain Captain of the Enterprise. (His decision??!? I won't spoil it for you!)
Its weakness (apart from being a first season episode) is that there are no big space battles or major alien encounters. But some smaller scale alien and ship things happen. And this episode is actually a great setup for one of the darkest (though sometimes still cheesy) suspenseful/ominous (though never paying off later) episodes of TNG, Conspiracy.
But before you go there, you can jump back to Datalore (to lead into future Lore episodes), if you like. Also Heart of Glory (for Klingon/Worf episodes), and Neutral Zone (actually, some say, a lead in to the Borg, but also, more obviously, The Romulans). Perhaps even Naked Now, Skin of Evil, and Encounter at Farpoint, if you're brave. But then follow this on in Season 2 with, at the very least, Q Who, The Measure of a Man, and a Matter of Honor. (And these all will effectively have formed mini-arcs.) Then you can watch (almost) everything from seasons 3, 4, and 5 without too much worry. And they will mean more when you watch them.
Anyway, long story short--my plan is somewhat higher risk, but higher return: if your girlfriend is anything like my 12-year-old nephews (huh??) get her to agree to give the show a proper chance by watching several episodes, then set her up with some of the more solid, early episodes, which you may both laugh at a bit but then she will be properly and emotionally invested when OMG Picard is Locutus! in BoBW, or LOOK Tasha is back??? in YE, or, oh no--IS SPOT GONNA BE OKAY???
