Ah, now this is a walk down memory lane... accompany the rewatch question with personal experiences of what, when, and how we
originally watched! I'll play...
I was too young to have caught TOS or TAS on their original runs. I discovered
Star Trek in after-school syndication sometime in 3rd or 4th grade. It stood out from the other options (mostly PBS kids' shows and '60s sitcoms), and I instantly became a fan. I remember the first episode I ever saw was "Catspaw."
Rewatch: I've watched every episode countless times over the years. I bought the DVD box sets that were released in 2004 (the ones in the plastic boxes, pre-remaster), and did a methodical production-order rewatch with my then-girlfriend. We did the same with TAS when it was released in 2006—my first chance actually to see those episodes, rather than just read about them. (We did the same sort of thing with box sets of Babylon 5, and Buffy, and The Wire and lots of other shows... not just Trek nor just SF.)
The movies: I've seen every one on its original theatrical release. I was still in grade school when TMP came out; high school for TWOK and TSFS; college for TVH and TFF; and law school for TUC. (During my middle school and high school years I was also a regular at the local library, and devoured the James Blish TAS adaptations, Bjo Trimble's
ST Concordance, and any other Trek-related reading I could get my hands on.) As the TNG films came out, I was gradually more disappointed with each. I remember going to see ST09 with friends in a state of excitement about seeing a fresh new take, and walking out in a state of shock and disappointment.
Rewatch: I have the box set of the special editions of the six original-crew films, and I'll pull one out to watch with friends (or occasionally solo) on occasion. I don't own any of the movies after those, nor have any desire to.
TNG I watched from day one. It was one of the very few TV series I followed in college (when you're a student you have better things to do). I wasn't alone in this; there was usually a big gathering in the dorm's TV room. Later, in law school, I had a couple of classmates who were also fans, and we used to gather at one of their homes each week to catch the new episode. (It was during those same years that I started accumulating the Trek novels from Pocket, courtesy of a local used bookstore that had a huge collection of them.)
Rewatch: I don't own TNG on DVD, nor any later series. I've never done a methodical rewatch. I'll occasionally go back and stream a particular episode on a whim (I just recently did that with "Yesterday's Enterprise," for instance), but I find that most of them don't hold up to that very well... the bad acting (especially Frakes) and mediocre writing (Piller-filler and technobabble) test my patience.
DS9 I also watched from the beginning. I was out of school by then. I kept with it until the late '90s, around the end of S5, a period when I (mostly) gave up on TV in general for a little while.
Rewatch: I've gone back to catch most of what I missed, notably including the final stages of the Dominion War. I feel like the show would reward a methodical rewatch, but I've never yet taken the time to do one.
VOY I also watched from the beginning, but never enjoyed, and gave up partway through S2, definitely before I stopped with DS9.
Rewatch: you must be kidding. How much would you pay me?
(Note: a later girlfriend was a huge fan of VOY, apparently because it's the first Trek show she discovered when she was a school-age kid. In particular she adored the Doctor — who really should have been given an actual name, if only to avoid confusion with a much more famous character who goes by the same sobriquet — and I agree he was one of the show's few highlights, but even so, we never attempted a rewatch.)
ENT I also watched from the beginning, but also gave up partway through S2.
Rewatch (after a fashion): years later (I don't recall exactly when), I heard through the grapevine that S4 had a different set of writers and was actually good, and discovered that to be true when I caught up with it via Netflix.
DSC: I've watch each episode as it's come out, the first couple with a friend and then (after he gave up) on my own. I feel like the season as a whole was a very mixed bag; it improved until mid-season, then slipped rapidly downhill again.
I've only watched each episode once. Even if future seasons show improvement, I can't imagine wanting to rewatch this one.
And that's my history with Star Trek. Long story short, I think the franchise was pretty great for its first thirty years, and much of that material justifies rewatching... and then took a nosedive, and most of what came after doesn't.