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 Star 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 (played out in the holodeck) 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 a race of a single individual, and is granted personhood. 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.