There's two potential start points really:
1)
Series One (2005) - this is when the show restarted. Who has been going for 50 years, but in 2005 came back after being off-air since 1989 (with a one-off special in 1996). This is the start of the modern show. It is a direct continuation from the old show, but gradually seeds in elements from the old series so you can learn all about the lore and such. This series introduced a whole new generation to Doctor Who in Britain who hadn't grown up with it on TV.
2)
Series Five (2010) - This is the one (also series six) with the cute redhead you mentioned. In 2010, the show top brass swapped out, and they created this as a jumping on point. Similar to 2005, this is a continuation (it literally picks up about 30 seconds after the cliffhanger ending of the last 'era') but also all-new, and seeds in the information you need to know gradually. That said, it is fairly reliant on things established before - there's characters and enemies that continue their plot lines directly from the 2005 series, because knowledge is assumed - whereas when 2005's series was introducing enemies from the old series, they had to assume there were 10 year olds who had never heard of these guys. You won't be missing out if you haven't seen 2005-2009, but you will find you'll enjoy certain things and changes they made more if you have that knowledge.
Also there's
3) which is "Soon" - if you don't want to or lack the time to marathon a ton of old TV - as it sounds like the current Doctor is fairly likely to be coming to the end of his tenure at the end of this year. The show bosses are staying the same at least through 2014, but a change of Doctor means we'll probably get another fairly clean break and jumping on point.
I would personally recommend Series 1 - it has aged pretty poorly, but is still brilliant, I think. If you're more fickle about bad effects and a few real clanger episodes, though, Series Five may be the best place to start.
Seven series' sounds like a lot, but it's only 13 episodes a series, plus around 10-11 special episodes from Christmas and other major events. On Netflix Doctor Who is split in two - old (pre 2005) and new (2005 on). IMO the best path is just to find the 2005 series and click play and watch all the way through to the present day. If you enjoy that, people can recommend the very best of old Who (63-96) for you to check out if you're so inclined.
If you watch, please come join us in
GAF's official DoctorI Who thread, as we all love being voyeurs and hearing reviews of old episodes from people experiencing them for the first time!