Balance of Terror, while more or less a subhunt style battle of wits, also introduces the notion that Romulans are an offshoot of Vulcans.
The Naked Time - since this is the best introduction to the depth of a lot of the characters, showing them when kind of drunk.
Balance of Terror: Exciting, even on the 20th viewing, and shows that Kirk isn’t the hero just because he is good at fistfighting and snogging aliens. Based on the movie The Enemy Below with Robert Mitchum as the captain (which was based on a novel.) The movie is great also.
Trouble with Tribbles: funny and clever.
City on the Edge of Forever: time travel, but mostly it is great because Ellison sets up a real emotional dilemma.
While I love Doomsday Machine, I’ll add one no one else has mentioned: Errand of Mercy. I like that because it shows Kirk’s weakness with a beautiful bit of irony. I won’t spoil it; this comes near the end and you can’t miss it, or Spock’s reaction. Anyone criticizing ST because it shows Kirk as the perfect unflawed hero should watch this.
Balance of Terror
City on the Edge of Forever
Where No Man Has Gone Before
The Trouble with Tribbles
Amok Time
Oh, absolutely. Didn’t want to give the impression that I was raising anything more than a minor quibble. Certainly even if I hadn’t previously seen TMP I would have know “Logical Vulcan” through cultural osmosis alone.
Still, I think it’s often the elements that non-fans know through cultural osmosis that call for address- since cultural osmosis gives only a two dimensional cartoonish representation. I really don’t know what distinction would be made between an emotionless Vulcan and an emotionless robot. I’m sure after watching some Trek and experiencing Nimoy’s portrayal firsthand I’ll “get it” a little better.
So, although I had no difficulty with Amok Time, I still think I n00b should be pointed elsewhere first. Happily, I will grant, that viewing Amok blessed me with a far greater appreciation for the Futurama episode Why Must I Be a Crustacean in Love.
Won’t get a chance to watch another episode until later this evening.
Don’t know whether or not my articulation of the minor quibble with the Amok Time suggestion might lead anyone to amend their list according to my belated detail of the kind of suggestions I’m looking for.
Good news is I don’t think I’ll have any problem sitting through five episodes at the least. My first viewing of a full TOS episode was a fun and enjoyable experience- didn’t feel like homework at all!
ETA: When was the whole Human Mother thing first addressed??? Seems like the story of Amok Time would have required addressing it if it had previously been established. Had that bit of history not been written into the character yet?
Go to the next episode on my list. Journey To Babel will answer all of your questions. 
You of course mean Joan Collins. Alexis Carrington was another character that she played.
Humor. It is a difficult concept.
Well, it’s not that they gave up. Gene L. Coon had written the script (under a pseudonym, Lee Cronin, I think) because he wanted to make the point that Gene Roddenberry couldn’t recognize a bad sci-fi script if it bit him in the ass. Unfortunately for Star Trek fans, he couldn’t.
O.K., in the OP I did ask that the suggestions of episodes be listed in suggested viewing order, but I’ve wimped out and decided to go with the suggested episodes but to order them chronologically- boring, I know.
Watched The Naked Time and Balance of Terror.
Even if every Trek fan in the Universe would place Amok Time in their “Best Episode” list, I think The Naked Time and Balance of Terror are better placed in the “Best Introduction” list.
I don’t anticipate becoming a superfan, but I’ve enjoyed these three episodes very much.
Still a little confused about Spock’s human heritage. It is mentioned in The Naked Time, so it was obviously established very early on. I’m surprised that it was treated as irrelevant- not even mentioned- in Amok Time.
Also, it needs be mentioned: 1966 Nichelle Nichols had an amazing body! Yowsahhh!!! I never knew. Wow. Just amazing. Growl!
Ooooh. I… I had no idea. I love you.
No love for the OP bringing it up to begin with???
I’m chopped liver?
The score is really something else though, isn’t it?
Oops! Sorry. Yes, of course, I love you too.
How about Devil in the Dark. She’s not just another pretty face.
Slightly O/T, but after recommending Balance of Terror in my own list, I rewatched it. Definitely one of the best. The idea of a nuclear war with an interstellar enemy we’ve never seen, and never do see for a hundred years after the war, is wonderfully creepy and haunting.
Too bad about that artificial gravity causing plaster chunks to fall on the Romulan centurion. That’s the kind of thing you have to overlook to enjoy TOS.
Wait until you see Mirror, Mirror.
Just five? Yikes. Eeep. Um… OK, then:
Amok Time - Interesting stuff on Vulcan society. Spock under stress. A very good look at the Kirk/Spock/McCoy dynamic.
Journey to Babel - Nifty diplomatic intrigue and spycraft. Spock’s parents and lots of angsty family drama. Kirk has to fool Spock and only barely succeeds.
City on the Edge of Forever - A very well-done time-travel story, with an agonizing dilemma faced by Kirk. Some of McCoy’s best scenes, both as an OD’ing maniac and as a soft-spoken country doctor.
Mirror, Mirror - A deliciously evil parallel universe and a canny, calculating, bearded Spock. Who could ask for anything more?
The Trouble with Tribbles - Very funny episode, and clearly the cast is having a blast. Some of Scotty’s best scenes of the whole series: “Well, I figured we were big enough to take a few insults, weren’t we, sir?”
Watch Mirror, Mirror. In the alternative evil universe the uniforms are skimpier.
Heh. If the OP want to look at more amazing bodies in amazing costumes, I humbly suggest the following fine episodes:
Mirror, Mirror - Nichelle Nichols, nuff sed
Who Mourns for Adonais? - how does that dress stay on?
The Cloud Minders - Spock is interested in her mind, really
What Are Little Girls Made Of? - a robot - a GIRL robot
The Menagerie - Orion slave girl, nuff sed
According to the actress (Leslie Parrish), sheer willpower and the weight of the cape. Everybody one set was scared to death it would somehow fail in action, but she felt quite comfortable in it.