TNG is a great place to begin. It's usually what I recommend to anyone who hesitates to start the original. It matures the series and gets into the philosophy of trek more.

You can always go back to TOS later if you really get into the whole universe of Trek. It's light-hearted and campy, the characters are cultural icons, and it's fun to see where everything got started.

Going to be that person and suggest the OP try Voyager.

It seems to be the most accessible of the older shows for younger, newer viewers. It was the most watched of all the classic shows on Netflix.

It covers all the classic tropes and provides endless fodder for memes here.

It’s uneven throughout its entire run, but also has some of the very best episodes ever. New fans really attach to the characters, and there’s no refuting that it unabashedly leaned into the weird.

Star Trek: Deep Space Nine is my favorite Star Trek of all time to this day. The setting on a space station rather than a starship whizzing all over the galaxy makes for a lot more development of characters and storyline than the more episodic nature of other Trek shows. Plus the chemistry the ensemble cast has is just fantastic, and you literally cannot get too much Armin Shimmerman.

A lot of people suggest TNG, but I actually recommend the TOS movies. Start at Star Trek II: The Wrath of Khan, watch the movies in order. You can skip the 5th movie without much loss.

Watching the movies will give you enough knowledge to understand why the original crew was such a big deal, even being referenced in universe later. It also gives you a basic understanding of augments, which do come back as later plot points.

Then watch through in order. TNG, skipping some of the junk episodes. Then DS9, then VOY. If you want to watch the TNG movies, I'd say skip them except for First Contact. First Contact is a bad Trek movie, but it's watchable and at least puts it's budget to good use. It introduces you to Picard's famous memed meltdown and it introduces the borg queen. If you watch it, and are going to watch ENT you need to watch the movie first because there is an ENT episode requiring having seen it to fully appreciate.

I personally like ENT quite a bit, aside from a few of its more egregiously bad or boring episodes, it has plenty of pretty good episodes.

Star Trek: The Next Generation (TNG) is one of the most popular shows in the franchise (and definitely meme'd to death online). With a stellar cast, interesting development of characters, and fascinating action and psychological stories, it helped build a modern setting for the franchise to grow into.

Also, it's one of the only series that received a modern sequel series (Star Trek: Picard). Sure, there are occasional crossover episodes between series, but TNG was so popular, they got most of the old cast back to reprise their roles decades later. So if you're invested in the characters of TNG, you get to see what adventures they're up to years after leaving Starfleet.

ALSO also... TOS and TNG are the only two series to get full-length feature films made alongside their respective TV shows. (Even the modern reboot films are set in the TOS timeline, following the characters from that show.) So if you'd like to see TV show characters on the big screen, TNG will give you those big budget productions along with your show.

The first TNG film (Star Trek: Generations) actually has Captain Picard running into Captain Kirk from TOS, who got stuck in some sort of time anomaly in space. It was awesome that they found a way to do a crossover between the two series, considering they take place almost 100 years apart from each other in the Star Trek timeline.

Fwiw, there was a "remastered" version of TOS, which updated many of the special effects. It's a lot less egregious vs Lucas' Star Wars remasters, but it's still a definite change. Some liked it, some didn't. But if the old effects and visual artifacts are putting you off, it may be worth a try.

My first show was The Next Generation after some half-hearted attempts at the original series. It was a fantastic starting place. It has interesting characters and entertaining storylines, and from what I understand it introduced the more modern Klingon design. If you end up enjoying it, it segues nicely into both Deep Space 9 (which, to me, is just as good) and Voyager (which has not caught my attention quite as well as the other two series).

TNG I'm is the series that really establishes the modern universe, and having seen TOS isn't really necessary. You could also start with the original movies instead of the show.

I'd start TNG, and whatever anyone might suggest DO NOT skip s1 or 2.

Everyone is different, but I generally use https://kethinov.com/startrekepisodes.php to guide my viewing experience.

Anything that is a 7 or better on the audience scale, I watch. Anything less than, but has a high personal score from the maintainer I generally give consideration.

It includes a lot of trivia and additional details that may help me get some of the cultural references I could otherwise miss. The episodic nature of the early shows mean you don’t miss much skipping around. If you finish a series and want more, go back and watch everything above a 6 and so forth.

There are some watch guides out there (but I haven't verified any myself, sorry!). If you do watch TNG, VOY, DS9, ENT there are many junk episodes that could be skipped. I've seen them all so I just skip manually when I see one of them start.