There are certain sort of "natural" starting point episodes, so take your pick:
1. The Eleventh Hour - First EP of reboot season 5; new doctor, writer, director, TARDIS, companions, the lot. Very natural.
2. The Christmas Invasion - First Season 2 EP, and first with the beloved 10th doctor. You won't immediately know the companions, however.
3. Rose - The first of the reboot. Obviously a very natural starting point.
4. Doctor Who - I'm referring to the 90's motion picture that took place between the classic and new series. It will introduce you to the feel of the show, as well as the Master. Plus, by the end, you will already have seen every 8th doctor appearance. (Besides NOTD)
5. Spearhead from Space - This is IF you are interested in the classic series. This is the first color episode, and the first 3rd doctor episode. It's another very natural-feeling, fresh starting point.
6. An Unearthly Child - Obviously, the very first episode ever made is always an option.
Hope this helped!
Start with "Rose" in the reboots or go back to the original "An Unearthly Child" which started it all.
A bit of trivia, the Doctor nearly didn't take off due to very poor ratings for the first episode. The reason? JFK was killed. The producers got the BBC to re show the first episode a week later and history was made.
Start at the 1960s version
The last one! Work backwards.
I know it's been on for an extremely long time, so I really don't think I should start all the way at season one. What season or Doctor should I start on?