Start at the beginning, with An Unearthly Child, the first Doctor Who story. As much for ‘historical significance’ as anything else. Then move on to the second story, The Daleks, because it introduces…the Daleks.
Tomb of the Cybermen, with the 2nd Doctor. I’d also recommend The War Games, but it’s a 10-episode story…I’m not sure that fits in with the idea of ‘skimming’
For the 3rd Doctor: Spearhead from Space (his first story and the introduction to the Autons), The Time Warrior (which introduces the Sontarans and Sarah Jane Smith), and Terror of the Autons (which introduces The Master).
The first story with the 4th Doctor: Robot. Also, City of Death (written by Douglas Adams, and a candidate for 'best Doctor Who story ever). Then skip to the end of the Tom Baker/beginning of the Peter Davison era for a trilogy of stories: The Keeper of Traken (which reintroduces The Master after a long absence), *Logopolis *(the last 4th Doctor episode), and *Castrovalva *(the first 5th Doctor story and a tribute to M.C. Escher).
*Earthshock *features the return of the Cybermen, and a rare occurrence: the death of a companion (Adric). Don’t miss The Five Doctors, which was the 20th anniversary special story. Then the last story with the 5th Doctor, The Caves of Androzani.
Nobody likes the 6th Doctor, but for completeness sake watch his first story, The Twin Dilemma. The Two Doctors is notable because it features a return appearance by the 2nd Doctor.
For the 7th Doctor, watch Remembrance of the Daleks (which is set in the same time and place as the first episode of the series), and Survival, which was the last episode of the original series.
It’s important to remember (or realize, depending on whether or not you ever knew in the first place) that the ‘classic’ Doctor Who stories were serials: generally four 30-minute episodes. So each ‘story’ is about twice as long as the ‘new’ hour-long episodes. So these 18 stories (not counting The War Games) will only take you a day and a half, more or less, of continuous viewing. It’s a good thing tomorrow is a holiday 