This is a great question. I like to recommend Data's Day for newbies because he sort of goes around and introduces every character, but it's a pretty unconventional episode.
I think I would go with "Doomsday Machine"!
That said, I also think Darmok would be a good fit too, the concept itself is unique in that as far as I know it's the first it was used in science fiction, but the manner in which the concept is presented "Enterprise crew encounters a problem that can't be solved with guns or technology" is pretty darn representative.
Good point! I forgot that it was mostly the two of them on the planet. It is also somewhat unusual in that there is no distinct B-plot. The events that take place in orbit are directly related.
"Darmok" is so far the only Trek episode I've shown my spouse, because she wanted to know what the deal was with a t-shirt I had with that episode's famous line on it. It did the job, she seemed to get into it despite being only vaguely familiar with the show.