Ideally, the place to start would be the original William Hartnell episodes from the sixties. Except many have been lost, and the show was in black and white in those days, more talk than action with really unconvincing monsters, cheap sfx, and wobbly sets.
The Patrick Troughton episodes were a little bit livelier, because the Doctor was a bit younger in appearance and played a recorder.
When he morphed into Jon Pertwee, the show switched over to color, and the stories became more Earthbound because the Time Lords (of which he is one) sentenced him to being trapped on Earth.
Tom Baker's Doctor had more freedom to travel around through time and space, and a lot of fans (including myself) consider him the best Doctor of the original incarnation of the series. He certainly made the role his own...and held on to it for the longest time.
When Peter Davison took over the role, I guess a lot of folks were expecting him to carry over the playfulness of the character he had played on "All Creatures Great And Small." Instead, his Doctor just blew around like a leaf in somebody else's wind from episode to episode.
Colin Baker tried to inject alienness back into the part, and he was the first actor to be fired from the role, rather than walk away from it of his own accord.
The show was cancelled when Sylvester McCoy had the role. It had just become totally irrelevant by this time, and people just stopped tuning in.
Paul McGann's Doctor only had one shot in a made-for-TV movie, which never became a series, but he is still officially the eighth Doctor.
Thus ended Doctor Who for a while.
When the show was revived in this century, Christopher Eccleston took on the role for one year, which established that the Doctor's race was now extinct. And rather than half-hour episodes that took about a half-dozen episodes to tell a single story, the new format was one hour, self-contained episodes, BUT there was also an ongoing mystery that would stretch across the entire year.
When David Tennant became Doctor number ten, he really endeared the character to a whole new generation, the way Tom Baker did in his time. It was a one-two combo of his charm and the kinds of stories being told. His exit was the most emotional one in the whole franchise.
Now that Matt Smith has the part, the series has taken a darker and much less warm fuzzy approach, but it's still as gripping as ever.
The general rule seems to be whichever Doctor you start watching...chances are that one will end up being your favorite.
I would tell you to start with the Eccleston episodes. The first one is called "Rose." While the revival series does maintain continuity with its predecessor, the producers knew full well that the current savvy genration of fans were going to demand the best music, sfx, acting, and stories. And they have consistently delivered. It was this season that introduced many elements that the show has returned to time and again. It will hook you, explain what has transpired previously when it needs to, and it will whet your appetite for what will happen in the future.
There are a couple of channels on JustinTV that show episodes online, but you are at the mercy of what they want to broadcast. I suggest watching the DVDs, because you get a lot of wonderful extras, besides.