Here's the guide on how to watch Dr. Who
Start with the first episode of the 2005 series. It kinda sucks, but just go with it.
Be aware of the following:
1) When he dies, he regenerates into a new body, hence the different actors. Eccleston, Tennant and Smith in the modern. In order, first season, then the brown-haired guy with the weird thing about glancing around and long pauses after saying elongated things like 'yeah...' during explanations and then the guy with the bigger chin. A quick image search will clear it up.
2) He often has companions. In the new series: (Eccleston and Tennent) Blond girl, (Tennent) black girl, angry redhead, (Smith) redhead who becomes part of a married couple, black-haired girl. Until Smith's run, the biggest concern with figuring them out is that the blond one loves the Doctor and the married couple (after they're married) have a daughter who makes a lot of appearances. If you want to watch Smith's run, start at the beginning.
The companions have storylines, but are less a concern for individual episodes, and the show is very episodic.
3) The character will talk about the Time War quite a bit. It was a war. In time. His planet got blown up. You'll be okay from there.
4) He has a lot of enemy races that show up. The things that look like pepper shakers are Daleks. They blew up his planet (sort of. He still blames them). There's a backstory and all that, but per-episode, accept that they're xenophobes who want to exterminate, and they'll tell you all about that.
There are also Cybermen as the other major recurring enemy. They're the androids. Also a backstory, but if you know that they're cyborgs and generally filling the 'evil' slot, you'll be fine to understand the episode you've found yourself in.
After that, start more or less anywhere, but I'd recommend the start of a Doctor's run or the start of a Companions, just to keep things connected and avoid jumping in. I thought the series got good at the start of Martha Jones' run (the black girl), but that's me. Eccleston had his moments and an excellent finale to the first season.