If you are interested in a quick and fun overview of the old series, check out the "Dcotor Who Years" videos on the official BBC Doctor Who website. These feature clips from every story (bar three stories early on from which no video survives) in chronological order, all set to a sountrack of songs popular from when the stories were first broadcast - from the Beatles in 1963 to Kylie Minogue in 1989. A running text commentary also will tell you what exactly you are watching and its significance in the history of the series (as well as some jokes - "fine accents" in The Gunfighters indeed!).
The one drawback is that the videos are pretty small, so you won't get too much detail. But if you are new to the series and are curious about its long history, this is a good way to experience a good general overview of it from beginning to end. The text does mention the periodic spoiler if you are totally new to the series (it reveals what the "shock" was in Earthshock, for instance), and there is one tiny spoiler for the new series at the start of the '70s video (but it's very minor and won't really spoil all that much).
The videos can be found here. Look under the "Classic Series" on the right side for the three "Doctor Who Years" videos (the Sixties, the Seventies, and the Eighties):
http://www.bbc.co.uk/doctorwho/video/index.shtml
And while there isn't much in the way of spoilers in these particular videos, there are plenty of them on the rest of the official site - so if you want to remain unspoiled, don't go wandering around much there!