I'm a huge life-long fan and have watched it all, multiple times, and love every episode, but let me give you are realistic list.
Here are what I recommend, in order. Don't start with the new series, start with the old, specifically:
1) Destiny of the Daleks (Tom Baker)
Watch this one first, it will get you hooked on to Doctor Who!
2) An Unearthly Child (William Hartnell)
You can see how it all began
3) Spearhead from space (Jon Pertwee)
Has a monster you will see later in the new series
4) Inferno (Jon Pertwee)
Today's Slashdot story about drilling to the core of the earth is of relevant interest, you will see what happens
5) Frontier in Space (Jon Pertwee)
One of my favorite stories with the Master
6) Genesis of the Daleks (Tom Baker)
See the Daleks
At this point look for the Key to Time series (Tom Baker, 6 stories), watch a few episodes by the 2nd, 5th, 6th, and 7th doctors, then you can move to the 2005 series.
2nd Doctor, watch War Games
5th Doctor I recommend Earthshock, the Five Doctors
6th Doctor watch The Two Doctors, Revelation of the Daleks
7th Doctor watch Remembrance of the Daleks
After this, watch more Tom Baker, or move to the 2005 series.
Enjoy!! You will love this show. It's truly the best science fiction show ever created.