“Let That Be Your Last Battlefield” and yes, it’s a 3rd Season stinker.
Season 3 is correctly described as sub-par, but I sometimes think fans take it to an extreme. First, the general quality is heavily skewed by a few true stinkers–the aforementioned Spock’s Brain right out of the gate, plus Spectre of the Gun, Plato’s Stepchildren, The Way to Eden, and the preachy Let that Be your Last Battlefield. However, these same fans often forget about equally bad (but far less frequent) outings from earlier seasons–like the heavy-handed The Apple, the Halloween-silly Catspaw or The Alternative Factor (if anyone can explain this one, be my guest).Second, there are far more “bottle episodes” where Kirk, Spock, and (often) McCoy are stuck together on the same junky set for most of the episode–The Empath, Requiem for Methuselah, Whom Gods Destroy plus some of the ones mentioned above. For an action/adventure/sci-fi show, confining the main characters like this is a huge liability for the story.
I think these two factors cause many fans to write off a lot of quality episodes from season 3. IMO The Paradise Syndrome, Is There in Truth No Beauty, For the World Is Hollow… and All Our Yesterdays stand up with the best episodes from the earlier seasons, and even the middling outings where they’re actually out doing something are no worse than, say, The Return of the Archons or The Gamesters of Triskelion (five hundred quatloos says you’ll agree).
Actually, as laughable as “Spock’s Brain” is, it ISN’T the worst episode. That honor belongs to “A Little Child Shall Lead Them”, with Melvin Belli, the [del]ambulance-chasing[/del] personal injury lawyer as the monster.
Here are 5 that i think someone who hasn’t watched the show would enjoy and make them possible want to watch more. I would also suggest you watch the versions with the updated Special Effects since people who did not grow up with the originals would find the cheeziness of them distracting I think. These aren’t necessarily my favorite episodes just ones I think a new watcher would enjoy and want to watch more.
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City on the Edge of Forever.
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Mirror Mirror
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Balance of Terror
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Space Seed
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A Taste of Armageddon
Not just fun for the great costuming (oh my), but really a cool ep. Helped explain the Kirk/Spock relationship, too.
And: “THAT was the equation!” was so exciting when I first saw it. 
Oh, I have no objection to cheesy special effects; I watch Doctor Who. (I think the versions on Netflix have the updated ones, though.)
Thanks for all the suggestions, guys!
I’m the perfect age, and watched TOS from the first episode during the first run. It was a different experience slowly learning the evolving Trekiverse.
My list, with commenatry
The Corbomite Maneuver: First non-pilot episode filmed, with a shouting Spock and different uniforms. First good McCoy humor - “If I paid attention to every light around here I’d wind up talking to myself” - as he is talking to himself. (Quotes from memory.) Shows Kirk’s strategic brilliance, and sets the tone of peaceful contact.
The Naked Time - not for half naked Sulu - though that the Japanese dude’s fantasy is French says a lot about Earth of the time - but for the back stories of Spock especially. See the increasingly frantic Uhura as the only sober person left respond to Sulu addressing her as “fair maiden” - “Sorry, neither.” Copied by TNG, not as good.
The Enemy Within - written by Richard Matheson, good psychological study, we get to understand Kirk.
Balance of Terror - introduces the Romulans, first real space battle, a great strategic duel. The movie it is based on “The Enemy Below” - a submarine movie starring Robert Mitchum - is worth watching also.
Tomorrow is Yesterday - really almost a comedy as the Enterprise goes back to 1969. Air Force guy to Kirk: We’ll lock you away for 200 years. Kirk: That will just about be right. I’m not sure it is a great episode, but I loved seeing the Enterprise in the skies of Earth. Plus, they got the day of the Apollo 11 launch right (Wednesday.)
City on the Edge of Forever- talked about plenty already. Note the jars of Vaseline smeared on the lens for each shot of Joan Collins.
Errand of Mercy - introduces the Klingons, explains why there is no war, and has a brilliant ending which shows Kirk as a flawed human.
Trouble with Tribbles - already mentioned plenty, but interesting in it is one of the first shows to make use of a plot point from a previous show. Now common, fairly rare back then. Great bar fight.
The Doomsday Machine - perhaps my favorite episode with a great problem, with William Windom giving Shatner chewing up the scenery lessons, with a great script by Norman Spinrad (even if not quite the Ahab in Space plot he intended), great music, an excellent solution, and some great lines.
As for the number one show to avoid - not Spock’s Brain, which at least has hot space babes, but “The Alternative Factor” from the first season which doesn’t even have camp value.
Close. It’s “And the Children Shall Lead.”
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The Tribbles episode
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The one where Kirk reads from the Eed Plebnista.
“The Alternative Factor” doesn’t even make sense stoned. I have no idea who green-lighted that episode, but they had access to some strong shit.
“Journey To Babel” has to be in the top five just for all the canon it establishes. Spock’s parents Sarek and Amanda, glimpses of several Federation races and virtually the only original mention of the Orions. Plus Spock being the quintessential [del]jackass[/del] “logical” Vulcan.
Plus McCoy gets the last word for once. 
I dunno. If it isn’t spoiled for you and you don’t try too hard to figure how it could be spoiled for you I think it has a pretty good gotcha.
More WTF? are people thinking social commentary than a riveting episode of Sci Fi drama but still.
And it’s like all those “gotcha” stories. It really only works if you didn’t see the gotcha and then only the first time.
And the Mud episodes are “cute and lighthearted” for lack of a better description.
Another noteworthy bit about this episode: the script was the first professional writing job for David Gerrold (who was only in his mid 20s when he wrote it).
Most of the good “must watch” episodes have been mentioned. The OP also asked for the “must watch” because they’re astoundingly bad episodes:
“Spock’s Brain” has gotten some mentions. You also shouldn’'t miss “The Way to Eden” AKA The Space Hippies episode.
But my all-time favorite so-bad-it’s-good show is “The Omega Glory”. It’s basically 40 minutes of Kirk and the villain (a Federation captain that’s gone bonkers) beating the crap out of each other with fists/axes/sticks, followed by the most jaw-droppingly corny, jingoistic and stupid conclusion in the show’s history. Seriously, if you don’t already know the twist at the end, avoid all spoilers and just revel in the stupid. I was 11 the first time I saw it, had no idea what was coming, and was cringing on the couch hoping no one would walk in while I was watching.
My opinions:
Must Watch
Season 1:
The Man Trap (because it was the first one aired)
Balance of Terror (because it’s excellent)
Shore Leave (because it’s fun)
The Galileo Seven (because it has some great Spock character development)
Arena (because Gorn)
Court Martial (good Star Trek courtroom drama)
Space Seed (KHAAAAAANNNN!)
Devil in the Dark (evil alien has actual motivations for being “evil”)
Errand of Mercy (the Organian treaty figures in many of the future Trek stories)
City on the Edge of Forever (one of the best)
Season Two
Amok Time (great Kirk/Spock character development)
Journey to Babel (great Spock character development, and we meet his parents)
The Trouble With Tribbles (one of the best)
Season Three
The Enterprise Incident (one of the few good ones in S3)
Don’t bother (only picking the worst of the worst here)
Season 2:
Who Mourns for Adonais (Greek Gods in Spaaaaace!)
The Omega Glory (about as subtle as a rock to the forehead)
Season 3:
Spock’s Brain ('nuff said)
And the Children Shall Lead (bad acting and washed-up lawyer galore!)
The Way to Eden (Hippies in Spaaaace!)
The Cloud Minders (only somewhat less subtle than Omega Glory, with more cheesecake)
Turnabout Intruder (laughingly sexist even by '60s standards)
Spock never wears a gangster suit in that episode. You’re thinking of “A Piece of the Action.”
I’m fond of that one too, if only for the hotness of the winsome blonde whom Kirk prevented from accepting her own execution. But for commentaries on the senselessness of war, I prefer “A Private Letter War” (aka “Vietnam in Space,” with its somewhat-atypically dark ending.
Barbara Babcock, later well-known for playing Grace, the hot-to-trot girlfriend of Sgt. Esterhaus on Hill Street Blues. Yup, she was a babe. 
Man, there’s a lot more time-travel than I thought there would be …
I have a soft spot for Changeling, since it was the first ep. I ever watched (grandma’s house after my grandfather’s funeral when I was 11).
I also see that Conscience of the King hasn’t been mentioned yet, like how the the plot of the play mirrored the plot of the ep. And the girl who played the daughter-without spoiling things let’s just say she nailed the role.