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

Mike using his YouTube money to buy and preserve old TV and Movie props is the best arc. Maybe in his retirement he can curate his collection

Ask me any other day and my answer will be different, just highlighting a few GOATs this tread is neglecting.

1: Raging Bull

2: Citizen Kane

3: Dr. Strangelove

Picard starts every day by broadcasting “Space… The final frontier” to the crew