There are some great answers here already, but as a huge Whovian I wanted to add a little more information.
As a complete beginner to the series definitely start with the 2005 Doctor Who series. The series will make perfect sense and there will be nothing revealed that will leave you confused (or at least nothing that won't be cleared up later in the series!).
There are old enemies/allies from the original series that will appear and you'll not realise it - but that won't affect your viewing experience in the least.
Now, if you watch the Series, like it and decide to consider watching the Original Series, consider the following:
Doctor Who kicked off all the way back in 1963. This is significant for a few reasons. The original series was in black and white. Now, I'm a child of the nineties and as much as I'm ashamed to admit it, a lack of colour really does affect a movie for me. I'm so unused to things being black and white that I find it much deeper to embrace older movies/shows. Of course that's not universal and I've enjoyed some fantastic old films and tv shows - but without question they require far more concentration on my part to really get into. If you're anything like me and you think that a lack of colour (at least for the first few series) would bother you, it's worth bearing this point in mind.
Unlike the original Star Trek, which was in colour and has since been touched up/remastered, the original Doctor Who episodes with William Hartnell do show their age. They're certainly watchable and enjoyable, but you'll need to concentrate and accept some very dark/grainy footage at times.
You'll also need to accept that as times changed, so did the character of the Doctor. Nowadays, we want a fun, energetic, brilliant, slightly sexy Doctor, which is where the likes of David Tennant (10th) and Matt Smith (11th) came from. Back then, the idea of the Doctor was that he would be a very intelligent, slightly dotty man. That's the character William Hartnell plays. An old, clever man, who is slightly mischievous. He by no means always saves the day. In fact, in many situations he's the one who gets them into trouble and his companions (Barbara and Ian, two teachers) are the saviours.
I'm bringing all this up because if you watch the new series, the character that the Doctor portrays is not how it was in the original series (at least to begin with).
Another issue to consider is that in the very beginning Doctor Who had an educational remit. This meant that for every "futuristic" serial it ran, it had to run a serious "historical" serial which would accurately tell a real life historical event (such as Marco Polo, the burning of Rome by Nero etc). Now this remit only last a few series, but it's another thing that's worth bearing in mind.
Finally, if you do decide to go back to the Original Series, there is one more thing you must accept. Due to the wiping policy of the BBC there are a number of episodes of Doctor Who that are missing (mostly Patrick Troughton, the 2nd Doctor's, but also a fair few of Hartnell's (1st Doctor)).
In other words, if you do start watching William Hartnell right from 1963, within a few years you'll start hitting episodes that you'll either have to listen to the audio from (they were broadcast over radio) or you'll have to skip entirely.
Having said all that, the big positive of going back to the Original Series is that you'll get to soak in all the Doctor Who mythology, hear, see, live and breath the universe presented and be a "true" Whovian! But - it is a long (and expensive) haul. Don't expect to finish anytime soon!
Unquestionably start with the Modern Series (2005). If you like it and want more and are considering the Original Series, consider the following questions:
- Will the black and white/grainy aspect of the stories bother you?
- Will the low production costs of original episodes bother you?
- Will the different presentation of the Doctor (not as strong/intelligent etc) as modern Doctor bother you?
- Will the fact many of the 2nd Doctors serials are missing, meaning you'd have to "skip" them bother you?
If the answer to any of these is yes, then perhaps the Original Series isn't for you.
If the answer to all of these is no, then start watching and enjoying :)