I'd say start with the reimagined series (2005-). The old Doctor Who (1963-1989) has its charms, but it's a very different product, with low budget visuals and a very didactic approach, so it's not for everybody.
The new series is a continuation of the old, but with modern sensibilities and storylines. The main character is a near-immortal Time Lord who can cheat death by "regenerating" his body, so eleven actors have played him so far: seven for the old series, one for a tv movie during the 90s, and three in the new series (Nine, Ten, Eleven). You have one season for Nine (Christopher Eccleston), three for Ten (David Tennant), and three (so far) for Eleven (Matt Smith).
It's an excellent show, I can't really recommend it enough.