What to Watch
If you're thinking of starting in the scary world of "classic" Who, then here is a list of what to go for in terms of beginning and Doctors.
For your very, very first classic story, I would recommend one of two. The Five Doctors, whilst not being the greatest story in existence, is good in that it introduces its viewer to many classic companions and villains, and gives a peek into what each Doctor is like. The other I would recommend would be Remembrance of the Daleks. Now, this IS one of the greatest Doctor Who stories, and feels modern enough so as not to alienate modern viewers. If you want to begin with one from each Doctor, however, then here are my recommendations
The First Doctor (1963 - 66)
There are three good starting points for the First Doctor - The Beginning Boxset, The Rescue/The Romans and The War Machines. Now, obviously, I'd go for The War Machines first, simply because it isn't a boxset, but for some of the best Hartnell stories, The Beginning is the place to start. It's also called The Beginning. What more could you ask?!
The Second Doctor (1966 - 69)
The main problem with the Second Doctor is that many of his best stories no longer exist (Google 'doctor who missing episodes' if you aren't sure.) The two best available Troughtons are therefore probably The Invasion and The War Games. These are both quite long (8 and 10 episodes respectively) so maybe seek out a copy of The Tomb of the Cybermen before having a go at these longer ones.
The Third Doctor (1970 - 1974)
The Time Warrior is a fun starting point for the first colour Doctor, but Spearhead from Space, The Three Doctors and The Daemons are also a nice place to begin.
The Fourth Doctor (1974 - 81)
For a Doctor who ran longer than any other, it's more difficult to pick out specific stories that nicely reflect their era. The Seeds of Doom and City of Death are both two of his best, but there are many, MANY more fabulous ones.
The Fifth Doctor (1982 - 84)
Earthshock seems to be held in strong regard by most fans, but personally I've never thought of it as particularly special. The Five Doctors is a good place to start, and The Caves of Androzani is similarly well regarded.
The Sixth Doctor (1984 - 86)
Poor old Sixie's era was troubled by BBC turbulence, but even so, there are some strong stories to be found here. The Mark of the Rani and Revelation of the Daleks are both good stories, the former being more 'fun' than the latter.
The Seventh Doctor (1987 - 89)
The aforementioned Remembrance is brilliant, as are The Curse of Fenric and Survival, the latter very much having a 21st Century 'Who' feel to it. Yes the cat is awful but you can look past it.
The Eighth Doctor (1996)
There's not really much to choose from, as he only did one film. Buy it as part of the Revisitations 1 boxset and your 4th and 5th Doctor needs will also be fulfilled.