I thought a little ironic misplacement of the comma would go a long way; but here's the situation.

My roommates and I were drinking last weekend, and I made a comment to one of them about a Picard facepalm. My other roommate was like "Who's Picard?". Because apparently in her entire lifetime she has absolutely zero exposure to anything Star Trek related.

This saddens me, because it essentially means her understanding of reality isn't even complete. I want to help her at least see the parts of Star Trek that have shaped culture as a whole, but I understand there is a lot of Star Trek, and I don't want it to seem like a torture session, in fact I would be happy if she developed a hunger for Star Trek in the process.

She works in theatre (live not cinema) and I've already concluded ToS would be a good starting point, since she would probably appreciate the campiness of the sets/effects and rawness of the acting. So what I really need is a list of the best ToS episodes (both title and number) and a good order to play them in.

By the best I mean ones that are iconic, that you couldn't possibly even hope to understand modern human culture without having seen. (I.e. Guardian of Forever, Menagerie, etc) so that as she's watching she can't help but think to myself "Oh my god, I understand what that crazy guy in line at the grocery store was going on about now." And also so that she can understand how it has helped to shape modern cinema and tv.

To answer all the troll questions already:

Yes, she's hot.
No, she's off limits.