Star Trek is absolutely the direction you should go. Don't save it for later, enjoy it for now!
TNG is of course one of the absolute best series. I don't feel like there's any real need for me to vouch for its quality. It's got Patrick Stewart as Jean Luc Picard, Brent Spiner as Lt. Data, Michael Dorn as Worf, and of course LaVar Mother F***ing Burton as Geordy LaForge. Plus you've got recurrent characters such as Guinan (played brilliantly by the one and only Whoopi Goldberg), the very trolly godlike entity Q, and the hilarious Lwaxana Troi (mother of Deanna Troi, the excellent Betazoid counselor). You've also got things to look forward to such as the Borg, and, just because it bares mentioning again, the appearances of Q.
Having said that Deep Space 9 (DS9) is probably my favorite of the series. While Captain Sisko will never be as good as Captain Picard, and Lt. Dax can't hope to be as awesome a science officer as Lt. Data, the show more than makes up for it with its more focused narrative. Unlike the Enterprise (which travels all over), DS9 is a stationary space station hovering above Bajor and not too far from the borders of the Cardassian Empire. It is, of course, also right next to the worm hole which connects the Alpha Quadrant (federation territory) to the Gamma Quadrant (ruled by The Dominion). You'll get a much better feel for the peoples who come from these various forces, and start to understand and love the region of space which DS9 inhabits. And it offers some excellent characters of its own. Captain Sisko is still brilliant, and in later seasons you'll find a familiar face in Worf (O'Brien from the enterprise is also on the station as the engineering chief, and he'll grow on you quite a lot). But you'll also get the station's shape shifting security chief Odo, who's a really fun character. And replacing the stoic bartender roll played by Goldberg is the station's hilarious money grubbing Ferengi, Quark and his maligned brother Rom, running their bar/gambling hall/holosuites (think holochamber but somewhat skeezier and rented by the hour). There is also the recurrent roll of the ex-Cardassian Obsidian spy and assassin Garack, who has taken the role of DS9's tailor.
Really either series would be an excellent option. I put off watching Star Trek for a long time myself, but when I did finally start watching it I simply couldn't put it down. If you give it a try, I'm sure you'll find the same.
