Do you want to start in chronological order?
Star Trek: Enterprise
The very first show in the Trek timeline is Enterprise. It starts at the foundation of the Federation when it was just humanity and the Vulcans working together in any real capacity. The Denobulians eventually came on board, as did various other alien races, but this is the beginning of it all. It features sci-fi legend, Scott Bakula, as the captain of the ship and runs four seasons deep.
The latter half is considered “better” than the first half but I think that’s a lie. While the overall final two seasons have a more interesting conflict, the first two seasons have the better individual episodes. It’s all fantastic, however.
Do you want the best most critically acclaimed?
Star Trek’s The Next Generation or Deep Space Nine
Critically acclaimed, or “most popular” are just fun buzz words that don’t mean anything. Star Trek, like all forms of entertainment media and art, is subjective. Yet, the two shows that most fans rave about are The Next Generation and Deep Space Nine.
TNG is the series with Patrick Stewart playing Jean-Luc Picard and has arguably the most eclectic core group of characters in the franchise’s history. You have a blind engineer, a robot (android) helmsman, a Klingon at a time when Klingons weren’t too friendly with the Federation, an empath, and a future time traveler. It’s wild
DS9 is both more grounded, and somehow not. At its heart, DS9 is about a father and son living and working on a Federation outpost, while helping the local government put themselves back together after an occupation. At its soul, it’s a tale of spirituality and what constitutes belief.