TenForward: Where Every Vulcan Knows Your Name

Watch two series intermittently

The traditional inculcation is to catch mid-series TNG episodes at random.

Hang on, I've got a past answer saved here somewhere. Yeah here we go:

Episodic television from the linear broadcast era means you can jump in damn near wherever and it'll be an okay time. Usually. Deep Space 9 has more ongoing storylines and every series has some episodes people just haaate. The Next Generation has half of its what-were-they-thinking episodes in the first season. Nonetheless I'd recommend TNG as the starting point, because it has the right mix of grandeur and restraint.

The Original Series is technicolor sci-fi horror dressed as episodic space exploration. It's campy and melodramatic and by god does it know it. Some episodes were blatantly "what's available on the back-lot next month?" and are a coin toss between stupid and incredible.

TOS had some movies (roman numerals) and they range from "what if an episode was three hours long" to "modern-day San Francisco zoo heist." At their best they're fun and at their worst they're even more fun.

The Next Generation is high-concept ethical debate framed as the experiences of naval officers. The cast is seriously talented and the writing is usually excellent.

TNG had some movies (bald guy on the poster) and they were written by people who didn't like the show for people who didn't watch the show. You have to turn your brain off, but they're well-directed.

Deep Space 9 was a direct response to Babylon 5. it's heavy on other species criticizing the openly optimistic and quietly human-centric Federation. It never goes full "revisionist western," but it gets dark in places. It was also a deeply indulgent excuse to explore characters and cultures from prior Trek.

Voyager did the opposite, by yeeting a ship to the ass end of the galaxy, where nobody's heard of the Federation. They're just trying to keep it together and beeline back toward home. It's the same formula as most TOS / TNG episodes, but with fresh character dynamics and new rubber-forehead aliens.

Enterprise is an origin story starring the guy from Quantum Leap, and the tone is just... weird. It's not awful, but I'm not personally a fan. Apparently it had an ending even more disappointing than Quantum Leap, which is fucking saying something.

JJ Abrams did some reboot movies replacing The Original Series, and they're everything you expect when reading "JJ Abrams did some reboot movies." Great casting, though.

Discovery sounded like it got up its own ass a bit. There's a point in any long-running franchise where the people making it grew up as fans of it, and you require strenuous effort not to create a feedback loop of shallow fanfiction stupidity. See: RTD-era Doctor Who, numbered Star Wars sequels.

Picard sounds like season one was written exactly like the TNG movies, which is pretty far from a compliment. Season two is apparently more of a 1990s Trek reunion and I still can't care.

Strange New Worlds sounds like it's doing everything right, as a revival of TOS-era "wagon train to the stars" problem-of-the-week storytelling. The universe is big and weird and terrifying and we're out there penciling in the map. Plus they pulled off a musical episode. That's always a good sign.

Honorable mentions to Galaxy Quest, an aliens-abduct-actors comedy that is accidentally one of the better Star Trek movies, and The Orville, Seth MacFarlane's extremely Seth MacFarlane love letter that's honestly a pretty solid realization of the formula and absolutely knows it. Also neatly illustrating the difference between parody and satire.

TNG, DS9, VOY, BSG