I started typing up a couple recommendations and then it just grew, so here's this! It might be useful?

The original series of Star Trek is a show that really existed before plot arcs were a thing. Occasionally you get references to past episodes, and the characters do evolve a bit over time, but in general they are all standalone episodes. You can skip around.

I picked three episodes i like a lot from each season and then bolded the one that i think is the most successful out of the three. In general, I think that the first and second seasons are the best. There are plenty of other favorites i have in there, but i had more trouble picking finding favorites from season three.

season 1:

• The Corbomite Maneuver - a really solid episode that shows off kirk's brain
• The Devil in the Dark - kirk and spock face a deadly blob thing
• The City on the Edge of Forever - this gets placed on top of a lot of lists. and bonus domestic kirk and spock

season 2:

• Amok Time - fuck or die featuring Spock
• Mirror, Mirror - evil alternate universe and badass uhura
• The Trouble with Tribbles - prob the ep id recommend anyone to watch first. super fun and easy to watch (for extra fun watch the Deep Space 9 episode Trials and Tribblations afterwards XD)

season 3:

• The Enterprise Incident - infiltrating a Romulan ship
• The Tholian Web- good storytelling, nice character moments
• The Empath- really focused on the relationship between the main trio

kirk backstory episodes:

• Shore Leave - kirk's old bully, and an ex girlfriend
• Where No Man Has Gone Before - his long time friend Gary Mitchell gets godlike powers
• The Conscience of the King - Kodos the Executioner
• Court Martial - Kirk gets framed, plus an ex girlfriend
• Operation: Annihilate! - kirk's brother
• Obsession - a creature that killed a bunch of crewmembers in kirk's past comes back. he doesnt deal well

There are definitely episodes that aren't great but are still fun to watch, but there are others that have questionable messages or are just painful. I'd say these are skip-able:

• Metamorphosis - 'okay so ive been sexing up this cloud creature buuuut its a lady cloud creature right?' also a comment along the lines of women crew being replaceable with other women. ew
• The Paradise Syndrome - white savior/noble savage nonsense. at one point the girl doesnt understand how kirk could possibly remove his shirt? reeeeallly uncomfortable
• The Enemy Within - while there are really cool things about this episode, it also features an attempted rape that isn't treated well in the script at all
• Turnabout Intruder - probably the most sexist ep of the show, and the reason my Kirk comes from before it happens. the message is 'women should just be happy with doing womens things'
• The Omega Glory - more noble savage nonsense with some added asian stereotypes. it tries to flip the book on racist attitudes but fails SO HARD. also kirk saves the day by reciting the declaration of independence. so bad.

"Mudd's Women" and "I, Mudd" feature a recurring super sexist character, and the episodes are...okay at refuting him, but not great. I'd say they're watchable episodes if only to see how quickly Kirk gets tired of him

If you're in the mood for silliness:

• The Naked Time - a thing lowers the crew's inhibitions, everyone goes a bit off the rails
• The Squire of Gothos - features a pretty ridiculous villain, kind of a proto-Q
• Catspaw - idek whats goin on with this ep. halloween things plus a realllly odd plot
• The Trouble With Tribbles - such a good
• A Piece of the Action - gangster planet
• Spock's Brain - SO BAD, but in an mst3k kinda way
• Spectre of the Gun - Wild West