I’d say a good place for a new viewer to start would be “Rose” - the first episode of the new series, which re-introduces the character & basic concepts fairly well.
If you want to delve into the original series, I’d also say that a good place to start would be the very beginning. But of course, the original series covered a loooong span of time, and (let’s face it) includes a whole lot of clunky howlers. So, as a long-time die-hard fan, I’ll chime in on what I think are the key stories of each Doctor:
Doctor #1 - William Hartnell:
“An Unearthly Child” - introduces the original cast, the TARDIS & basic series premise.
“The Daleks” (also known as “the Mutants”) - Introduces the most enduring alien / villain in the show’s history.
“the Aztecs” - a historical adventure that actually holds up very well, all things considered.
“the Dalek Invasion of Earth” - the second outing of the Daleks, and one of the all-around best early serials.
“the Time Meddler” - The first time another Time Lord apart from the Doctor (and possibly his grand-daughter Susan) is shown. (Even though they weren’t called “Time Lords” yet.)
Doctor #2 - Patrick Troughton:
It should be noted that the BBC junked a lot of the master tapes of episodes from this period, therefore a whole lot of stories are lost or incomplete.
“Tomb of the Cybermen” - the earliest Cyberman story in which all episodes are still intact. It’s also reputed to be the best early Cyberman story.
“the Mind Robber” - a rare surreal story in which the Doctor & his companions travel to the Land of Fiction.
“the War Games” - Not an especially good storyline actually, but it is essential viewing if only for the final episodes in which the Doctor’s back-story gets filled in, and the Time Lords are officially introduced.
Doctor #3 - Jon Pertwee:
“Spearhead From Space” - First 3rd Doctor story, and essentially a second pilot, as it introduces a whole new format with an Earth-bound Doctor aiding a military organisation known as UNIT in fending off alien invaders.
“the Silurians” - one of the best of the UNIT era stories, featuring one of the most intriguing “alien” races.
“Terror of the Autons” - The main storyline is essentially a retread of “Spearhead”, with the same alien race invading again, but it does introduce two important characters - the Master (the Doctor’s Moriarty/nemesis) and popular companion Jo Grant.
“the Daemons” - Pertwee’s personal favorite DW story.
“the Three Doctors” - a rare story featuring the meeting of all three Doctors in action.
Doctor #4 - Tom Baker:
“Robot” - Again, a lackluster main story, but it introduces the fourth Doctor, so watch at least the first episode.
“the Ark In Space” - You’d swear that the writers of the movie “Alien” saw this story before they wrote that screenplay.
“Genesis of the Daleks” - Perhaps the single best Dalek story ever. Introduces one of the best ever villains - Davros.
“the Deadly Assassin” - The only story from the original series in which the Doctor has no companion character. Also gives a lot of detail about his home-world - including crucial information about the process of regenerating.
“the Sun-Makers” - another surreal story that parodied the British taxation system. Also like it because it excellently show-cases Leela, his companion of that time.
“the City of Death” - the episodes that scored the highest ratings ever for the original series. Interesting use of time travel in the story, cameo appearance by John Cleese, and has one of the best companion characters in it - Romana (2nd incarnation.)
“the Keeper of Traken” - the first part of a loose trilogy of serials that ressurected the character of the Master, and lead up to …
“Logopolis” - Another regeneration story. Second story of the trilogy.
Doctor #5 - Peter Davison:
“Castrovalva” - Third in the trilogy of stories.
“Kinda” - another oddity, a sci-fi tale based on Buddhist/Hindu philosophies.
“Mawdryn Undead” - Reunites the Doctor with the Brigadier (from the UNIT stories), introduces a nebulous new companion Turlough, and another good story about time travel conundrums.
“The Caves of Androzani” - a surprisingly gritty story (given the light-hearted tone of the original series.)
Doctor #6 - Colin Baker:
“Vengeance On Varos” - This is a terrific story, but alas it’s just about the only good one from this Doctor’s era. The series was seriously sloping downhill during the mid-80s.
Some might say that the “Trial of a Time Lord” is important to the overall canon of the series, but that doesn’t excuse the fact that it’s a long, boring slog that’s painful to watch. All you really need to know is that the main villain of it - the Valeyard - is revealed to be a possible future (evil) incarnation of the Doctor, and the Time Lords are revealed to be a decidedly shady, self-serving group rather than the benevolent, wise overlords they claim to be.
Doctor #7 - Sylvester McCoy:
I had given up on the series by the time he took over the part. I’d say “Remembrance of the Daleks” is an OK story, but overall the series seemed generally tired, with little life left in it.
The FOX TV movie is the one & only television adventure of Doctor #8 - Paul McGann, who I liked in the part. Alas, everything else about this outing is unwatchable.