I suspect you will find the newer stuff more accessible. I would start with the Christopher Ecclestone season and watch through to the present day.
After that, you may want to sample some of the older stuff, so here are a few suggestions for the various Doctors:
William Hartnell: An Unearthly Child, The Daleks, The Dalek Invasion of Earth. The first two are included in a set called In The Beginning
Patrick Troughton: The Tomb of the Cybermen
Jon Pertwee: Spearhead from Space, The Sea Devils
Tom Baker: Robot (his introductory episode), Genesis of the Daleks, Pyramids of Mars, Robots of Death, Deadly Assassin, City of Death
Peter Davison: Earthshock, The Caves of Androzani
Colin Baker: Revelation of the Daleks
Sylvester McCoy: Remembrance of the Daleks
In my opinion, seasons 12-15, the first four seasons of Tom Baker, are the best of the older series. But the list above should give you a taste of the whole thing. See what you like.