I would suggest watching starting with the 2005 rebirth of the series. They made it expecting you to not have watched the original series.

From then on there is a lot of overlap and continuation, some more subtle than others, so you could just watch one of the doctors, but part of what makes it special is the wealth of possible collisions so you may not need to have seen the reason why such-and-such character perks up when they hear the Doctor coming, but it's what made me appreciate the show.