I actually don't have a satisfactory answer to my own question.
Star Trek was a prominent part of the popular culture when I was a kid, so it was easy to have a basic knowledge of TOS without actually seeing it.
I think I probably saw Star Trek IV first, and the first episode I can clearly remember watching is "Unification II", though I'm pretty certain I saw something before then.
This is probably well beyond the scope of what these nights are about, but I do wonder what they did with the "Picard" season three bridge set.