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