After Roo attempted to troll Star Trek nerds with a loaded question, he was eventually inspired to actually watch the movies, or at least the statistically good even-numbered ones. I offered to write up my recommendations on which ones to watch for someone pretty unfamiliar with Star Trek.
Inspired by the self-aggrandising moniker "Machete Order" for the Star Wars movies, I'll call this the cackhanded journey. Unlike Machete, I'm not trying to reshuffle things to preserve a big reveal. I'm just recommending that you avoid watching certain films because they're not worth your time, and giving you a bit of optional background TV research if you're new to Trek.
The Original Series
The Motion Picture
Paramount's original efforts to reboot Star Trek morphed into a movie, and we got The Motion Picture.
First, watch the episode The Changeling. This has a similar plot to the film, so it is interesting to contrast the low-budget TV treatment of the Nomad probe (a meter-long metal box with some lights inside) with the cinematic treatment of Vejur (a 21-meter model representing the 78-kilometer wide vessel, itself at the centre of a projected energy cloud described as being "over 82 AUs in diameter" — 12 billion kilometers, or twice the average diameter of the orbit of Pluto).
Then watch The Motion Picture. It is not a great film, but it is worth watching at least once if you've never seen it before. It's more sci-fi than the action "Wagon Train to the stars" of the series, and is about the most accurate realisation of what Gene Roddenbery wanted. It also made much less money than Paramount expected and nearly killed Star Trek. Fortunately Paramount went ahead with a sequel anyway, pushing Roddenberry to one side.
The Wrath of Khan
Before the movie, watch the episode Space Seed to learn who Khan is and why he might be wrathful.
Meyer's movie is a tight action flick, and deliberately much more like the series in tone. In my opinion it is the best of the Star Trek movies, and one of my favourite movies ever.
Although I still can't get past the obvious "two-dimensional thinking".
The Search for Spock
Before the movie, watch the episode Errand of Mercy, which introduced Klingons to the series. Although we've already seen Klingons briefly in the opening of The Motion Picture, they feature more heavily in this film. Don't worry about them missing forehead ridges in the series.
The Search for Spock is a filler movie at best. It exists solely to reverse two events from the previous movie, and get rid of two extras that didn't need to be gotten rid of. Those characters could just have not appeared in the later films. Especially given the original actress didn't return.
But it's worth watching once, maybe even twice if you are a fan of Christopher Lloyd.
The Voyage Home
Star Trek is not without humour, so in the next of the even-numbered "better" films we have a fish out of water (pun definitely intended) film where the main characters get to be out of their depth (somebody stop me).
If you are bored and have too much time on your hands, you could watch the episode Tomorrow is Yesterday which introduces the slingshot manoeuvre used in the film. But it's not a great episode. (You could also watch the other time travel episode Assignment: Earth to see something bizarre, a backdoor pilot in a Trek episode.)
The Final Frontier
Don't watch Star Trek V, just … don't. It commits the sin of introducing a sibling never before mentioned, has a boring plot, and everyone starts behaving stupidly.
Instead, you could watch the episode Where No Man Has Gone Before. This very early episode features an energy barrier at the edge of the galaxy (a plot point repeated in the film, except it is now at the centre of the galaxy, because Trek fans definitely don't care about details).
But you must definitely watch Shatner of the Mount, which is a music remix made from one of the DVD extras. It is sublime, and the only good thing to have come from this atrocity.
The Undiscovered Country
Before watching The Undiscovered Country, you could watch more Klingon episodes. Friday's Child (Julie Newmar!) and A Private Little War are okay. The Trouble with Tribbles is another lighthearted episode, and worth seeing before you watch all of Deep Space 9. Which you definitely should do, it's the best.
As a bit of a whodunnit (although the who is very obvious) and a bit of a space battler, this is one of the three best Trek films. And you've not experienced Shakespeare until you have read him in the original Klingon.
The Next Generation
Before watching Generations, watch the Next Generation episode Relics which explores the ideas of the distance between generations, and that of handing over from the previous to the next.
Generations isn't awful, but it also isn't that good. It is an episode's worth of plot stretched to a movie length, but it's worth watching once. Especially to see Shatner's last scene, given he doesn't ham it up as you might expect.
If you don't know who the Borg are, you could watch the episode Q Who that introduces them. You should watch Best of Both Worlds, parts 1&2 to see the Borg properly take on the Federation (and a familiar looking nebula), and Descent to see a more Data-centric take on dealing with the Borg.
This is an absolute cracker of a Star Trek film with great action sequences. I was always slightly resistant to the introduction of a single avatar to represent the hive-mind, but I guess that's futile.
I suppose she was for those less familiar with Trek to be able to focus on a single villain doing nefarious things rather than an amorphous, abstract threat of "being replaced". Because Invasion of the Body Snatchers was a terrible film.
If you still have time on your hands and like the character Data, you could spend a couple of hours watching Insurrection. But just the once.
As the exception that proves the rule, Nemesis is a bad even-numbered film. Ignore Nemesis. It is basically poor fanfic (eek a spaceship with many much lots weapons compared to the Enterprise; a previously unknown sibling, of sorts) and has no redeeming features.
If you are a fan of JJ Abrams, you might watch the first two of his reboot films. But they're different in tone, take a lot of liberties with what I consider to be the spiritual core of Trek, and are little more than average action-comedy flicks.
The third one is pretty boring, destroys the Enterprise (a repetitive event in the films) simply to strand people, and uses hip hop music as a weapon.
In redeeming features, I think Zachary Quinto portrays being Vulcan better than Nimoy did, Benedict Cumberbatch is the best villain, and Karl Urban is great as Bones. And they're far better than either Final Frontier or Nemesis.
However, I don't like that Kirk goes from being a capable and compassionate leader to being a selfish child who basically trips and lands in the Captain's chair. Don't @ me.