So, if I know nothing about Dr. Who, but wanted to give it a shot, where would an aficionado suggest I begin the journey?
You can start with the relaunch from 2005 with Christopher Eccleston. That's all on Netflix. It start's with the episode "Rose". It was designed so you can start there, but if you're familiar with the old show you catch stuff.
If you wanted to start old school without wading through lost material, you can start with Jon Pertwee's run in 1970 with "Spearhead From Space". He was the third Doctor, but you get a brief synopsis from supporting characters. The previous season ended with The Doctor changing and all companions leaving, so it was a fresh start. Last time I checked that story is still on Hulu Plus.
If you want to start all the way back at the beginning, "The Unearthly Child" was the pilot episode in 1963. Tracking stuff from there can be difficult, because there like 100 episodes that are gone.
Hulu actually has a lot of classic Who. But if you're impatient, I suggest starting with the 2005 relaunch.