Last year I introduced a friend to Trek by starting with Wrath of Khan. She knew I was a serious Trekker and I often described to her what Trek in it's various incarnations is about and what it means to me and many other people around the world, which she found kinda interesting. Her initial idea of Trek, from hear-say AND mixing it up with multiple franchises, has been that it was more of a childrens show with a lot of *pew-pew* and funny Aliens.
After finishing Wrath of Khan she was obviously surprised that it was a rather serious movie. Themes like getting older, revenge, friendship and loss didn't skip by her, and she seemed to like the crew.
So she was open for more (the inception worked) and next were ST III, IV and VI, over the time of two weeks. No need to rush things.
Right now we 're mid-season seven of TNG and mid-season two of DS9. It's become something of our routine after work where we watch one or two episodes of each series together. She really likes both crews, but I think she actually prefers Deep Space Nine now, because she loves the Odo & Quark dynamic, likes the Ferengi (!) and is really looking forward to when Worf gets stationed at DS9 (Worf is her favourite on TNG). And she's always on the look-out for Morn and asking how he can afford to spend so many hours at Quarks and if he doesn't have a job or a wife!

Also, she works as a personnel coach, so she is obviously intrigued by the differing but effective commanding styles of Picard and Sisko, she really looks out for how both manage and handle conflict situations.