I would recommend The Next Generation or Deep Space 9 first.

The Original Series is pretty campy and low-budget. I don’t think Star Trek really became Start Trek (as we know it) until Roddenberry had a second crack at it, which was The Next Generation. Be warned, both of these series start out pretty rough.

The first 2 seasons of TNG is a little corny, although they have some very good episodes, like one planet they visit which is a matriarchal society, and they show the male-female gender roles reversed. In my opinion, their 1940s nostalgia episodes really did not age well, fortunately they stopped doing that in season 2. Dr. Polanski gets a lot of hate, but I thought she was a fine character. I think, season 3 of TNG was by far the best, with some of their boldest political statements. One episode, Picard is accidentally seen as a God by a pre-industrial society, in another, Commander Data is judged by Starfleet to be a person of his race of a single individual. In yet another episode, Mr. Data creates a child for himself which Starfleet deems to be their property, and results in one of my favorite Picard quotes of all time: “Hold your ground Mr. Data. (To the admiral) Order a man to turn his child over to the state? Not while I’m captain.” By season 6 and 7 they clearly start running out of ideas though.

In my opinion, Deep Space 9 was the hardest to start, but probably the best series of all I have seen. The first 17 episodes were so boring I could barely get through it. But starting with Season 1 episode 18 (titled “Duet”) and every single episode after it was consistently very good. They also, I think, make many more, and much bolder, political statements than The Next Generation ever did.