Your New to Who FAQ: A Guide For People Ready to Meet the Doctor
The beginning of the new series is where you should start. That's "Rose" starring Billie Piper and Christopher Eccleston as the Ninth Doctor. It was custom-designed to bring the series to a new audience after almost two decades of silence with only a poorly-received TV movie coming between the cancellation of the classic series and the start of the new.
Almost all of the new series is on Netflix streaming, save for a couple of Tenth Doctor specials from 2009 and the last five episodes. If you're stuck with Comcast on-demand services then you can pick up at "The Impossible Astronaut." That should be more than enough to get you caught up to the current season. You may get a little lost, but the thing about Doctor Who is just sort of going with it. Trust me, you pick up everything very fast.
I don't usually like science fiction...
Honestly, despite all the time travel that's going on there is not really a whole lot of science fiction. Sometimes it's straight horror, sometimes it's a mystery, sometimes it's a kind of fantasy, and most of the time it's a lot of running away from monsters. If you prefer all your TV shows to be hardcore realism like CSI, then maybe you won't like Doctor Who, but they really do explore a whole lot of different genres and never get bogged down in a formula.
What's the best episode to show someone that has never seen Doctor Who?
A lot of people will tell you "Blink" because it's a brilliant episode that requires little to no understanding of the mythos to enjoy. To me that's like trying to introduce someone to Pink Floyd by playing them Rasputina's cover of "Wish You Were Here." Sure it's genius, but it has little to do with the core of the art. The Doctor is barely in the episode, and there's not ever really another one like it.
If your friend has 90 minutes worth of patience I suggest "The Empty Child/The Doctor Dances." It's a compelling period piece that shows off Nine at his best, introduces Captain Jack, and has a great ending. You could also try "Planet of the Dead" if you own it (not on Netflix). It's one of the few one-off specials that doesn't end with a companion dying and is a killer adventure to boot.