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).