Recently a few people that I know who have no real history with Star Trek have started either dabbling in Trek or voicing an interest in it. Brian from CoolandCollected.com was just on our podcast and talked about his experience with the Star Trek movies. Aaron from MovieHodgePodge.com also recently did a review for each of the original movies.
If you're new to Trek and looking to get into it, the movies might seem like an attractive place to start, although it may not be the best place. On our podcast, Jeeg recommended jumping into Next Generation or DS9 first. I would concur, but probably favor NextGen as a starting point. The problem with the movies as both Brian and Aaron cited is that they seem to gloss over explaining things for new fans. Not only are things as basic as the relationships between the characters glossed over, but also the minutia like what's the scoop with cloaking technology and why can't the Federation use it? While I like that those things are not hammered home again and again in the movies for fans like myself who already know that stuff, I can see where there is a disconnect for new viewers.
I also think that the movies can seem a little confusing from a thematic perspective. The stories run the gamut from cerebral sci-fi to tense naval showdown to wacky time travel adventure to political thriller. And there's Star Trek V in there, too which is its own extreme. So I thought I'd make a list of six episodes (one for each of the movies) of the original series that a new viewer could watch that not only cover some of the small details, but also expose the difference in tone that the original series could convey. Presented in episode order...
The Corbomite Maneuver, season 1, episode 2
The Enterprise encounters a mysterious object in space that seems to be of a higher level of technology. Kirk and crew must outwit their opponent. Often times, Star Trek: The Motion Picture is compared to the episode entitled "The Changling," but "The Corbomite Maneuver" is a better version of that plot.
Tomorrow Is Yesterday, season 1, episode 21
This is not only the first big time travel story on Trek (although they had used the story-telling device already before this), but this is also Trek's first foray into humor. This serves as a perfect introduction to Star Trek IV and even uses the slingshot effect for time travel.
Space Seed, season 1, episode 24
This one is so obvious I contemplated not including it and going with something like Balance of Terror which was similar thematically. However, this episode establishes Khan and crew and explains why they were on Ceti Alpha V. Watching this before seeing Star Trek II isn't imperative, but certainly helps.
Amok Time, season 2, episode 5
Not only does this clear up some of what Saavik is talking about on the Genesis planet in Star Trek III, but this may be one of the best Kirk/Spock friendship episodes.
Journey to Babel, season 2, episode 15
This introduces Sarek, Spock's father, who appears throughout the movie series. However, this show serves as a great pre-cursor to Star Trek VI with it's political plot mixed with a whodunnit.
The Way to Eden, season 3, episode 2
Space hippies take over the Enterprise to go on a search for Eden. This is not a good episode, but watching this will give you a taste of what the inferior third season of Classic Trek is like and why Star Trek V could be argued to be pretty faithful to Trek in its own way, just not the preferred vision of it.
This doesn't answer every question or introduce every element but it's a start. If you like this sampling, then you could certainly go back and watch the whole series or look for other lists that are more aimed at being the "best of" type lists.