TNG and DS9 are the only two series that I really know well enough to make these recommendations for.
I would highly recommend watching TNG before DS9, even if she likes DS9 better. TNG and DS9 work best as companion shows, with TNG building up a lot of DS9 plot-lines in its latter seasons. I would also highly recommend against starting with any episodes involving Q or the Borg, or any of the series “highlights”. Let those plots build, and you don’t want to spoil some of the best episodes in a sampler. My go-to TNG introduction picks are:
-Cause and Effect
-Darmok
-Measure of a Man
DS9 can be harder to pick introductory episodes for. Its more serialized nature means that a lot of later episodes have a lot of spoilers. I would highly recommend against picking any episodes that highly involve the Dominion or the Dominion War, let that plot build itself up. That does however, leave us with a lot fewer options. My DS9 introduction picks are:
-Duet
-Past Tense
-Bar Association
Yeah I stopped doing it in High School after realizing that it’s some North Korea level bullshit. Got a few other kids in my homeroom to stop too, which really angered our teacher. She was a military spouse and would actually yell at us for refusing to participate. In the end, we compromised by standing but not reciting it. Was the begining of my political and social awakening.