Honestly what got me into Dr. Who was that Epic Rap Battles of History Doc Brown VS Dr. Who (Actually if you don't mind, it's just the Doctor!) and this Women of Dr. Who special they ran on BBC and I kinda got a little sample of 'nuWho' and it sort of whetted my appetite, I do remember watching some of Eccelston's run and a couple of Tennant episodes on PBS - but thanks to the magic that is Netflix - I decided to start with the 10th Doctor's first episode and go from there. I'm in the 4 series now and I have to say within the past month I have reeeally gotten into it.
My only lament about nuWho is... the seasons are so short! I guess I'm used to Star Trek or Stargate's like 23-22 episode seasons, so only getting 12-14 is kind of like "aww man it's already the finale?" And I know Tennant's time is almost done. But I do think something like Netflix is a great way to get a new watcher interested. You can hit up a random 'good' episode or just start at the get-go and go fromt here. I was going to start with Matt Smith's first episode, but I felt like I really needed to know more about the previous series before I could appreciate it better. I dunno why I skipped Eccelston's season, but I do recall watching maybe most of it already years ago, so I wanted to see something different before revisiting that.
I haven't watched any of the Classic Who straight through, besides a few clips on YouTube. So that may be my next stop - or check out Torchwood.
I've been looking for a new series to get into since I've already watched all the Star Trek seasons, SG1 and BSG so Dr. Who was a natural and at least it's still being produced so when I finish the back catalog, I can still look forward to new stuff!