I started watching last Monday(24/6)and we are actually watching the eps in air date order(starting with "The Man Trap"), something I now regret. C'est la vie, we've started and it's too late to back out now.
I would not have started someone out with "The Man Trap." It's kind of a downer and short on excitement. But the first season is bound to have something they like.
Depending on their personality, maybe "Shore Leave" and "This Side of Paradise" will click.
Or "The Corbomite Maneuver" and "Balance of Terror."
Or "The Conscience of the King" and "City on the Edge of Forever."
Or something else.
There is going to be a type of episode in there that's perfect for any open-minded, intelligent person who isn't prejudiced against the vintage age of TOS. And then they might open up a little more and accept all types of episodes.