In this list of suggestions, I wanted to showcase episodes that try to give a good overview of the cast and what they do and the feel of the series they are in. Note that these are what I feel are representative and relatively newbie-friendly, not necessarily the best episodes or even my favourites. So in that context, here are my picks and why:
TOS: “The Doomsday Machine” - it showcases the Enterprise encountering an alien menace, how the crew and ship works together to defeat it, the strength of the relationship between the central trio of Kirk, Spock and McCoy and it’s a great, exciting story to boot, with one of the strongest guest stars they ever had.
TAS: “Beyond the Farthest Star” - this is one which I felt they could have done for TOS if not for the budget. It feels the closest to a live-action TOS episode.
TNG: “The Best of Both Worlds” - especially the remastered HD movie cut. There really is no substitute for introducing people to TNG. This is the show firing on all thrusters: suspenseful, thrilling, and truly epic. It’s a cracking good story, pretty self-contained, moves at a steady pace, it has great characterisation and action, and if effective, it should lead to a lot of questions about the broader universe - the Borg, Guinan, Data - you can use as a means to lead newbies to more episodes.
DS9: “Civil Defence” - this may seem like an odd choice, being a bottle show, but DS9 was always a bit of a bottle show. I like this because it’s strong on character and shows off the relationships between the characters in a crisis situation, and it brings in both Garak and Dukat, whose interactions are always a delight.
VOY: “The Equinox” - another two-parter, but this shows off the difference between a principled and unprincipled captain and crew, a glimpse into how Janeway and Voyager could have succumbed to the dark side. (very close runner up: “Timeless”, just for an example of how VOY loved to muck with time travel and alternate timelines)
ENT: “Dead Stop” - a strange space station, Mayweather actually gets something to do (he gets killed), the need for the NX-01 to get repairs in unexplored space, and a genuine horror mystery to solve.
DIS: “Stormy Weather” - an episode so Star Trek it couldn’t possibly be more unless it put on pointy ears and attended a convention wearing a I GROK SPOCK t-shirt. It’s sciencey, it’s hopeful, it’s philosophical, it has kisses to continuity and even has everyone in the ensemble have a moment. And apart from that it is paced well, suspenseful and thoughtfully written. Well done all around. Honorable mention: “Magic to Make the Sanest Man Go Mad”.
LD: “wej Duj”- which I would argue is the finest Star Trek episode of the last few years and one of the best ever Star Trek episodes ever for the way it portrays the lower decks of Klingons, Vulcans, Federation and even Pakleds and… well, wait till the end credits. It also introduces fan favorite T’Lyn to the audience.
SNW: “Memento Mori” - there’s a lot to like about the episode - the submarine warfare conceit in Star Trek space battles has always been a signature so it’s nice to see it crop up again, and you don’t even mind that the correspondence is so blatant because it’s so well executed and the keeps the tension up pretty much throughout the episode.
I’m skipping PIC because that’s not really a show for fresh eyes.