As we approach series 8, a friend asked, "So, how do I watch Doctor Who? Do I start from the beginning? Where is the beginning? What do I do??"
This is a common question which I myself asked once, and getting an answer (from a Whovian friend) helped not be overwhelmed by over 50 years of episodes. So here is my take on it.
I'm assuming that the intent here is to watch past episodes to get a feel for it and understand context. You could of course simply start with the new, 8th season and move on from there, but there is so much good Who to watch! Also, Doctor Who builds on itself quite a lot, and, while usually not required, it's a lot more fun to get the references.
First of all, a quick summary of terminology and broadcasting history: Doctor Who was first broadcast in 1963 (with, obviously, the First Doctor), and went on somewhat steadily until the 90s (with the 8th Doctor appearing in a one-off movie). We'll call it "the original series", or "the old series". It then went on a hiatus for about 10 years. In 2005 the series resumed, with the 9th Doctor, and has been going on uninterrupted since. This is "the new series", or "the 2005 series".
Note that, in the Doctor Who universe, it's all the same Doctor. Same storyline and everything. The 2005 series was not a "reboot", not a "remake", not a "reimagining". It's just new episodes of the same series.
So, my general advice is: start somewhere in the new series until you've watched the whole of it it. Then, if you like it, go back and watch the old series (which has quite a different pace, and not everybody gets into it).
Having said all this, there are a few different options of specifically where to start, depending on your level of commitment:
Low commitment, or "Let's see what this is about"
If you just want to watch one episode and see what the fuss is about, I recommend watching "Blink" (10th Doctor, episode 3.10 of the 2005 series). When I first watched it, I played it again as soon as it was over :)
Alternatively (or following that), you can watch the double episode "Silence in the Library / Forest of the Dead" (10th Doctor, episodes 4.9 and 4.10). The final scene is slightly silly, but overall it's excellent.
More alternatively (or following the two above), watch the 11th Doctor's first episode, "The 11th Hour" (episode 5.1). It works particularly well as an introduction, as both the Doctor and the production team are new.
Finally, even more alternatively, and to mention an episode with the 9th Doctor (remember, the first one to appear on the new series), you can watch "The Empty Child / The Doctor Dances" (episodes 1.9 e 1.10). An excellent sequence, with an excellent - and, in my opinion, underrated - Doctor.
At any point in this sequence (including at the end of it), you can decide you like Doctor Who after all, and jump straight to one of the following:
Small marathon, or "Let's see one Doctor from beginning to end"
If you're in the mood for a more uninterrupted marathon, start with the 11th Doctor (series 5), and watch it all the way through until season 7. Remember not to skip specials, including the 50th-year one.
Alternatively, start with "Silence in the Library" (episode 4.9, mentioned above) and watch it from there on. This means a few episodes at the end of season 4, a few specials that comprised the 2009 "series", and then the 11th Doctor. I love this final sequence of the 10th Doctor's episodes.
Watch it all the way to the end, then possibly any new episodes of series 8 that might have already aired (with the new 12th Doctor), then go back to 2005 and watch all that's left, in order.
Full Who Marathon, or "I want it all!"
If you think you want to watch it all anyway, and are willing to forgive a period of some inconsistency as they adjusted into the new series (but even then with excellent episodes, and, again, in my opinion an excellent Doctor), just go with series 1 of the 2005 series. Watch it all, rewatch a few, memorise lines, from then on it's up to you :)
Now, if you'll excuse me, after all this Who talk I feel like I have a marathon to start...