A… Fascinating question.
I’d say that the key is to NOT start with the movies. A movie is a two-hour investment and all the movies require some kind of understanding of the “back story” of the characters and the universe… Yes, even the latest film Star Trek. Though (and some fanboys are going to yell and throw things at me for saying this) the JJ Abrams Star Trek is a great intro to the series and characters for new viewers – that was the intent of it, after all…
Have to say, the best way to introduce Star Trek is with classic Trek, and with one of two episodes: City on the Edge of Forever or Balance of Terror. The first, one of the best episodes of the series, and the second, a very well-written and acted production that is quite gripping and entertaining. Both episodes are character-driven, which is one of the reasons Trek is so well-regarded: the best Star Trek is about characters we care about going, to coin a phrase, where no man has gone before…