It took 14 years, but I finally got my spouse into Trek a while back with TNG. We are in the process of wrapping up season 6, and the question of where to go next has come up several times. The two obvious next steps:
- TNG movies
- DS9
And even then, there's chronology/production order. All the movies, then DS9? DS9, then the movies? Mix the movies into DS9? (Yes, I do realize that we "should" be mixing TNG and DS9 at this point if I'm going to bring chronology into the mix lol.)
But I'm still very much interested in getting her to watch TOS, if nothing else than for the base of knowledge. IMO it still holds up, save for some societal byproducts of its time. And love it or hate it, the remaster made going back a lot more palatable for people who might otherwise struggle to enjoy 60s effects. And from there it would be either TAS (which I have but haven't seen yet) or the movies... which flows nicely into Generations. As a bonus, all of this has been released on Blu-ray and is sitting on my shelf, and doesn't require paying for official SD versions of DS9 etc. (or setting up for viewing by unofficial means).
As for me, I never finished DS9 or Voyager... by the time the latter came out I was in high school and basically stopped having time to watch tv on a broadcaster's schedule. And then I was working nights by the time Enterprise came out... and then didn't start any of the newer shows because I wasn't caught up. ᕕ(ᐛ)ᕗ (So no spoilers please, same with opinions on the newer shows. It's been beat to death lol.)
tl;dr Keep on trucking or swing by the old haunts first?