As a TOTAL beginner, I'd suggest you start with the 2005 reboot, and watch them in the order they aired. It's both easier to get a hold of, and more modern. Plus once you're caught up, you can easily watch the new episodes as they're released.
In addition, if you're enjoying the modern Doctor Who series, I'd recommend watching Torchwood as well; preferably in the sequence it aired, as there are tie-ins with Doctor Who.
However, if you subsequently wish to watch some of the more "landmark" classic episodes, I'd suggest you go for 2nd, 3rd or 4th doctor stories first; specifically those involving the Daleks or the Cybermen. The 4th Doctor's Genesis of the Daleks is a fantastic story, and introduced a recurring villain (who appears in the modern series), and the 3rd Doctor's Master storyline introduced the character (who also appears in the modern series), so is well worth checking out. On top of this some of the 2nd Doctor's stories are well worth watching; specifically the early Cybermen stories like Tomb of the Cybermen, and the 2nd Doctor's final story, War Games.
I'd agree with other commenters, though, that the modern reboot is a lot easier to get into than the classic series, as the classic series is a bit dated in places, and the low budget special effects are often a bit corny. However, if you can deal with that, and enjoy the modern series, then I'd highly recommend checking out some of the classic series as well.