Read a summary of Star Trek: The Motion Picture (actually if you can, read a summary of a summary, the pacing of the movie is so ponderous). The even numbered movies are the best, but it's probably worth watching the 3rd one as well for continuity. The majority of the fandom pretends that the fifth movie never happened, and is well worth skipping.
The Next Generation (TNG) and Deep Space Nine (DS9) are the two that seem to have the strongest following with Voyager being a bit divisive among the fandom (depending on who you ask it either sucks or is good).
You will have to suffer through the first season of TNG and just trust that it will get better.
DS9 can stand on its own fairly well, but if you want continuity you can watch any time after season 5 of TNG. DS9 enters a fairly significant tone shift around season 4, and season 1, like TNG, is not necessarily representative of the entire series.
[edit] Here's a super-short summary of the above:
TOS
Movies: 2,3,4,6
TNG: at least first 3 seasons, if you don't like after season 3, drop. If you're impatient, you can watch just the pilot than skip to season 3 and not miss too much.
DS9: Seasons 4 and 5 of TNG develop some of the races that are prominent in DS9, so it's best if you wait until at least season 5 of TNG to watch, if you really didn't like season 3 of TNG, then this will stand on its own though.
Voyager: It was not my cup of tea; some fans liked it though.
There are certainly episodes that suck, but the better ones are things that you will find have informed our pop culture lexicon, and are referenced in many other things. Watch it for the characters, more than the individual shows, IMO.
Yes it is slow. But space is big and the slow pace gave the characters time to bond again and work-out the puzzle. I prefer the movie to the later ones because there is no fire-torpedoes quick-solution which wraps-up the story in time for the ad break...
The characters started off awkward and cold but slowly rediscovered who they really were, just as V'Ger did.
The Next Generation (TNG) took a few seasons to really hit its stride, some will recommend starting at season 3 - though I personally recommend suffering through the first two seasons at least once. Like TOS, TNG is highly episodic is mostly devoid of major story arcs. My favorite series, but probably more for nostalgia than anything else. "Best of Both Worlds" is a fan favorite episode, though I will recommend "The Inner Light" for anyone - even someone who has never even heard of Star Trek.
Deep Space Nine was a bit controversial when it came out, as it moved away from Roddenberry's idealistic "Wagon Train to The Stars" model of a show and more into conflict and gritty geopolitics set in the same era as TNG. There's a story arc and the characters experience far more development throughout the show than the cast of TNG ever did. I've come to enjoy it more as an adult than when I first watched it on the air as a kid.
As for Voyager or Enterprise, I haven't really watched either of them.
Voyager and Enterprise aren't as sequential as DS9 but should be watched in order. There are a significant number of episodes in each you can drop from your watching order, but I couldn't tell you which ones. They both have very low points in my opinion.
As for the movies I'm sure there's a watching order. Some of the movies are very meh. If you look online you might find a good order, but if you get really into just watch them in order.
The Motion Picture (#1) is basically an elongated TOS episode. Doesn't really influence any of the other films. Some enjoy it, some find it dull.
The Wrath of Khan, The Search for Spock and The Voyage Home (#2-4) form a trilogy and must be watched in order.
The Final Frontier (#5)...I loved it when I was nine. Skippable otherwise.
The Undiscovered Country (#6) should be watched after #2 through #4. It and The Wrath of Khan are typically tied for the favorite Trek film.
https://www.reddit.com/r/startrek/wiki/where_should_i_start
For DS9 in particular, this link was helpful for me:
https://www.reddit.com/r/DeepSpaceNine/comments/1fkijc/ive_m...
There are 3 episode lists in there and a user named Algernon_Asimov has a "Hmph!" comment with a link to detailed episode lists for TNG + DS9
Dive in, have fun, my personal suggestion is any episode that deals with time travel or the mischievous life form called Q. You can't go wrong there.
For movies: see the Original Series movies Star Trek II, IV, and VI (watching the series first not required), see Generations once (watch some of TNG first), see First Contact (watch most of TNG first), see Insurrection (surprisingly good), skip Nemesis entirely.
If you have to pick one series to start with, I'd say either DS9 (as a marathon) or TNG. Note that you'll get a very different flavor from both; TNG presents a lovely idyllic future with occasional hiccups, while DS9 presents a darker and grittier 24th century, for better or worse.
Enterprise is probably the hardest. Very RPG-like how they operate to their self made rules in what compared to later series is extreme limitations. Life is a lot different when the fastest transport ship flys at warp 1 and everything isn't deus ex machina away with AI or technobabble. On the other hand they overdo the "wooden ships and iron men" thing a little and there is too much "pew pew" action.
Next up is DS9. Some technobabble for dramatic license but still reads like a paper and pencil RPG fiction (which is good). Some "crewman of the week" cheesy soft filler episodes but mostly hard science fiction story. Has a reasonably hard science overall story arc. Arguably the best overall series. Has the best "hard" written "soft" character in all of trek, the Cardassian "tailor". He's not really a tailor, hope this isn't too much of a spoiler. Best sci fi on TV, maybe.
TOS is all over the map. Sometimes they did a good job of both hard and soft sci fi at the same time. Then there's "space hippies" or "spocks brain" or "spock needs to get laid" which we'll pretend never happened. There is an unrealistic (by hard sci fi standards) fixation on alien planets being the wild west desert. You NEED to watch certain individual episodes, like the doomsday machine.
TNG is the softest series, its General Hospital soap opera, on a space ship. Toward the middle either its a short story arc, sometimes halfway decent, or "crewmember of the week special story". The Wesley character is kind of the series Jar-Jar and you'd do best to skip episodes where he is "crewman of the month". Also the romance episodes read like fanfic from /r9k/. The alien civilizations are bad caricatures.
Generally across all the series from a hard sci fi viewpoint if an episode involves space nazis, time travel, Ferengi, romance, or a holodeck accident, it can be ignored with little to no loss with very few exceptions.
They maybe hadn't quite found their footing yet.
Real problem is the standalone episodes were not good
But imo, star trek TNG with Captain Picard is the best one. Deep space nine is good but is a bit slow, I think TNG is easier to watch.
I'm from the mid eighties and can't watch the original, it's too dated for me. So take that for what it's worth.
Other than that I'd start with The Next Generation and go in chronological order.