I think Star Trek Next Generation is the best one, with Picard... But maybe because I grew up with it.
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I have just started watching Star Trek a few years ago and I would agree that TNG is a good starting point.
I'd honestly advise watching TNG and then DS9. Give Voyager a shot, although I find it's a bit of a slog. You can always watch the first 6 movies as well. The other movies would be worth waiting will after TNG.
For the new stuff watch Discovery first and then Strange New Worlds (Or at least wait until the end of Discovery season 2 before starting SNW).
I would also recommend Lower Decks despite it being a cartoon. It's well worth watching.
I'd also recommend watching a few TOS episodes, at least The Trouble with Tribbles (both because it's good and that one DS9 episode)
Honestly it's worth watching the whole TOS just as a primer for the TOS movies. It didn't run for that long, not like TNG and it is entertaining if campy. The Wrath of Kahn and Voyage home are absolutely recommended.
The way I did it was to watch it all from the beginning and go straight through. It took a little over a year, but I wasn't going at the most steady pace. For The Original Series, you can find lists of "the 30 good ones," or something like that. Then I'd watch the first six movies. Then straight through The Next Generation, then the next four movies. Then Deep Space Nine, Voyager, and Enterprise. I'm in the camp that the J.J.Abrams movies don't count, so watching those is up to you. After that, the new ones! Discovery, Picard, Lower Decks, Prodigy, Strange New Worlds. It's a long journey but well worth it.
Optionally, many many people are willing to give you sample episodes from each series as a starter point, then you can just jump around as you wish. The only ones you really need to watch fully in order are Deep Space Nine, and the new ones as they're pretty much all serialized, or have significant underlying continuity.
Everything is a lot of Star Trek! (but you should really watch everything eventually) I recommend tackling it one series at a time, and it will go by faster than you'd think,
My wife and I started with Star Trek: The Next Generation (TNG) and I think it is a great place to start!
After that you could watch Star Trek: Voyager (VOY) or Star Trek: Deep Space Nine (DS9). DS9 being one of my favorite pieces of media ever. I love it so much. 🥲
Or if you want something more modern you can start with Star Trek: Strange New Worlds (SNW). The second season airs in a few weeks, and I hope it's just as amazing as the first season!
As someone currently on that journey, I’m thinking it’s well worth it! I started with TNG since that’s what I grew up watching and I’ve made it as far as the last season of ENT. Everyone talks bad about VOY and ENT, but honestly I think they’re great too in their own ways (although TNG will always be the goat in my book).
I’ll also add that part of what got me so hooked on it was listening to the podcast The Greatest Generation along with it. The extra running jokes and silly headcanon stuff really added a fun extra layer to everything! Helped me get through some of those rough early episodes too.
I'm late to the party, but I'll try and convey some idea of what to expect from "classic" Trek, so you can judge for yourself.
The Original Series—First off, you have to appreciate that this series was made before science fiction was really thought of as more than Flash Gordon serials. Yes, The Twilight Zone precedes it, but it is one of the major cultural touchstones in the legitimacy of the genre. This also means that it's filled to the brim with growing pains, awkward characterization, and yes, a fair bit of sexism. Indeed, there's an episode where the premise is that women can't be Starfleet captains (which, incidentally, was retconned away in Enterprise). I said all these things to temper your expectations, because while there are rough patches, there's some real gold here, including a legendary episode written by Harlan Ellison.
The Next Generation—Every subsequent Star Trek started out cowering in its predecessor's shadow, and that's why so many people are so hard on the first season. It actually features a straight up remake of an original series episode. But dear lord does this series grow into one of the greatest things to ever happen to syndicated television. "The Best of Both Worlds" followed up by "Family," and later on, "The Inner Light," will leave you in tears. Anyway, the through line of the series is that we can live up to our better natures, and understanding can triumph over division and aggression.
Deep Space Nine—Starts out having trouble escaping the shadow of The Next Generation, the first episode actually opens during "The Best of Both Worlds" and has Patrick Stewart as a guest star. Also has a rough first season. The Federation encounters the Anti-Federation and war breaks out. The most famous episode is unbelievably dark (called "In the Pale Moonlight", a reference to Tim Burton's Batman), but my favorite is a two-parter, "Home Front" and "Paradise Lost," where the chilling cliffhanger is Starfleet security patrolling the streets of Earth. Not doing anything nefarious, they're just there. The greatest fear on Earth is a militarized Federation. No matter how far DS9 treads into the darkness, there's still hope.
Voyager—Frankly, just a weird show. Voyager is sent to the other side of the galaxy and have to make their way back. The premise is fundamentally wasted: They don't need to worry about limited resources. The crew is a combination of Starfleet officers, actual terrorists, and a holographic Emergency Medical Ham. All this is ignored by the end of the third episode. Instead what you get is an adventure with Time Police from the Twenty-Ninth Century, serious defanging of the Borg, and planets of the week. In any case, the series is fun.
Enterprise—What an odd mish-mash. I guess they wanted to get away from the technological power creep of Voyager, but for some reason that turned into them being harangued by Time Police from the Thirty-First Century. The premise is that the NX-01 Enterprise is alone, since its warp drive is orders of magnitude faster than anything else Earth can field. This is wasted for the first two seasons, but actually kind of works in the third. Really the third season is the only thing worth talking about: the Enterprise has to stop an apparently malevolent coalition of races from destroying Earth, and as I said, they're the only ship fast enough to get anywhere. While The Next Generation has plenty of stories about diplomatic tensions with the cardassians, the danger from the xindi is much more present—even if you know everything has to work out, since the show is a prequel.
Now for the movies...
The Original Series—This is the first six movies. A lot of people like to hate on the first and fifth. The first is interesting, like the plot is virtually nonexistent, but it can absolutely be appreciated as a surreal experience. The second deserves all the good things people say about it. Pay attention to the fact that William Shatner and Ricardo Montalban are never onscreen together, yet their rivalry is thick and terrible. I think the third is the weakest. The fourth is a ton of fun, with Scotty trying to tell an Apple Macintosh what to do by talking into the mouse. The fifth is very strange, but it features the best scene in any of the movies so it's okay in my book. The sixth movie is also just fantastic, a striking allegory for the paranoia preceding the fall of the Berlin Wall.
The Next Generation—Nowhere near as good as the first six. Generations is an incoherent mess. First Contact is a silly action movie. Insurrection takes a good TNG premise ("Who Watches the Watchers") and makes it schlocky. Nemesis is just bad.
Everything I've seen after this point has been a dismal failure.