There's really two starting places for someone interested in Doctor Who: the first Eccleston (ninth Doctor) episode, and the first Smith (11th Doctor) episode. Each of them is something of a clean slate, only referring to past mythology in a way that newcomers will still be able to understand. Starting with Tennant (10th Doctor) is a bad idea, because you need to have seen Eccleston's season to understand what's going on in a lot of his stories, especially early on.
As for the classic series, whether you should seriously get into it or not honestly depends on whether you're willing to pay for it or not. They sell it as individual stories, instead of as entire seasons, so it gets very expensive very quickly. I grab classic Who DVDs every time I find one cheap, but starting from the beginning is /way/ out of my price range without resorting to piracy. The good news is that that's not really necessary to enjoy them. The show was more episodic than the new one, in the sense that you can enjoy most stories on their own merit, not needing to know much backstory beforehand. They were also more heavily serialized in that the stories all consisted of at least two episodes, usually consisted of three or four, and in one case, actually took up a full thirteen episodes to tell. That doesn't really matter for the DVD collections though, because like I said, it's sold by the story, not the season.