TOS: This might be the hardest to pick - maybe “The Corbomite Maneuver”? It’s fairly well-known, but is crowded out in a season full of classics.
TAS: Pretty much everything besides “Yesteryear” is underrated, but if I have to pick one, I’d pick “The Magicks Of Megas-Tu” for sheer weirdness.
TNG: “Remember Me” - really good episode. I think it’s pretty well-regarded, but it should be better regarded.
DS9: I’ve always been very fond of “The Sound of Her Voice,” for some reason. I don’t think I’d ever seen anything quite like it when it originally aired.
VOY: I think “Counterpoint” is the finest episode they ever did, but people rarely talk about it.
ENT: Uh…I guess I’ll say “Daedalus” - it does a lot of worldbuilding surrounding the transporter, but isn’t one of the big “lore” episodes that people tend to talk about.
DSC: “Su’Kal”. The cause of the Burn is something that speaks to the humanist core of Star Trek in a way that many people seem to overlook.
PIC: “Remembrance.” The first episode of the series is nearly perfect.
LD: I don’t know if the nature of the show lends itself to this exercise, but I’ll say “An Embarrassment of Dooplers”.
PRO: “Crossroads” - the kids arriving at a turning point, and making the kinds of dumb decisions that kids make.
SNW: “Ghosts of Illyria,” I guess - this series is still pretty fresh, but this is one of the better episodes that isn’t a “gimmick” episode.
Hypothetically, I think I’d structure it somewhat similarly to season 5: a couple of episodes to establish the stakes, two or three more to gather information or materials, then a couple to put the plan together and enact it, and then ideally at least one to catch up with Zora after the events of “Calpyso.”
As it is, there are so many unanswered questions surrounding this mission that I hope we at least get a novel or something out of it.
I agree with the general consensus here so far - every second Doctor seems to have a pretty good jumping-on point, so you could start with the 9th (“Rose”), 11th (“The Eleventh Hour”), 13th (“The Woman Who Fell to Earth”), or 15th (“The Church on Ruby Road”). The first episodes of the 10th and 12th could probably work, too, but not as well in my opinion.
I would probably go with “The Eleventh Hour,” but that’s because I prefer the storytelling sensibilities of Steven Moffat, who was running the show at that time - the show also looked noticably better by that point than it did in 2005.
But there are no wrong answers - if you get hooked, you’ll end up watching it all anyway, mark my words.