Maybe it's because this is where I started (as a youngin' back in 1980)... Start with "Pyramids of Mars", which is a great primer episode. If you like it, then watch The Android Invasion. If you can stomach the production values of THAT, you can watch any old Doctor Who episode with impunity! Then move onto "Brain of Morbius" and "Seeds of Doom". Now that you've gotten a decent dose of Tom Baker... go watch "The Five Doctors". This will get you a solid basis with the other characters and Doctors. "The Three Doctors" is optional, IMO unless you really like one of the first three.
But the best way to start is to watch the first and last episodes of each Doctor from 1-7, (read: 14 episodes) then start the modern series with Christoper Eccleston.
I also believe these are episodes ANY Whovian needs to have seen, (besides the first and last of any Doctor):
William Hartnell (Doctor #1): The Daleks, The Dalek invasion of Earth
Patrick Troughton (Doctor #2): The Tomb of the Cybermen
Jon Pertwee (Doctor #3): Ambassadors of Death, The Time Warrior
Tom Baker (Doctor #4): The Masque of Mandragora, The Invasion of Time, Full Circle/State of Decay/Warrior's Gate
Peter Davidson (Doctor #5): Black Orchid, Arc of Infinity
Colin Baker (Doctor #6): Trial of a Time Lord (I'm probably going to get killed for this, but if you watch it all in the same week and in order, it's one of the best stories ever written for DW!)
Sylvester McCoy (Doctor #7): Ghost Light