The policy in the thread is to spoilerbox stuff until a week after it has aired, to be fair to people who aren't watching things "live".
Doctor Who first premiered in November 1963 with William Hartnell as the First Doctor, and ran until December 1989 with Sylvester McCoy as the Seventh Doctor. This time period is known as "the classic series".
Paul McGann appeared as the Eighth Doctor in a TV movie in 1996. Long unavailable, this was recently released on DVD as "Doctor Who: The Movie"
Christopher Eccleston appeared as the Ninth Doctor when the BBC and executive producer Russell T. Davies resurrected the show in 2005. The numbering for episodes and seasons was restarted, so Eccleston's first series was released as "Doctor Who: The Complete First Series".
There are three usual starting points that come up in the thread:
- The Complete First Series: As previously noted, Eccleston's first (and only) series as the Doctor, and the first series after the show was brought back. Starts out patchy, but improves considerably after "Dalek".
- The Complete Fifth Series: The most recent series of the show, the first with Matt Smith as the Doctor and the first with Steven Moffat as executive producer. NB: While you don't have to watch Series 1-4 of the "new" show to enjoy the fifth series, I'd like to recommend you watch "Blink" from Series 3 and "Silence in the Library/Forest of the Dead" from Series 4 for Maximum Enjoyment.
- From the beginning: the hardcore approach. Substantial chunks of the show are missing forever due to BBC policies, and there's a lot of material to take in through your eyes. Furthermore, the formatting and tone of the show has varied quite a bit over the years.
"Is this how time normally passes? Really slowly and in the right order?" [The Episode List]
The Complete First Series - Christopher Eccleston as the Ninth Doctor
Give me the full play-by-play on the new series and the various spinoffs.
1.01 - Rose
1.02 - The End of the World
1.03 - The Unquiet Dead
1.04 - Aliens of London [Part One of Two]
1.05 - World War Three [Part Two of Two]
1.06 - Dalek
1.07 - The Long Game
1.08 - Father's Day
1.09 - The Empty Child [Part One of Two]
1.10 - The Doctor Dances [Part Two of Two]
1.11 - Boom Town
1.12 - Bad Wolf [Part One of Two]
1.13 - The Parting of the Ways [Part Two of Two]
NaN - Doctor Who: Children in Need (AKA: Doctor Who: Born Again)
The Complete Second Series - David Tennant as the Tenth Doctor
2.00 - The Christmas Invasion
2.01 - New Earth
2.02 - Tooth and Claw
2.03 - School Reunion
2.04 - The Girl in the Fireplace
2.05 - Rise of the Cybermen [Part One of Two]
2.06 - The Age of Steel [Part Two of Two]
2.07 - The Idiot's Lantern
2.08 - The Impossible Planet [Part One of Two]
2.09 - The Satan Pit [Part Two of Two]
2.10 - Love and Monsters
2.11 - Fear Her
2.12 - Army of Ghosts [Part One of Two]
2.13 - Doomsday [Part Two of Two]
The Complete Third Series - David Tennant as the Tenth Doctor
3.00 - The Runaway Bride
3.01 - Smith and Jones
3.02 - The Shakespeare Code
3.03 - Gridlock
3.04 - Daleks in Manhattan [Part One of Two]
3.05 - Evolution of the Daleks [Part Two of Two]
3.06 - The Lazarus Experiment
3.07 - 42
3.08 - Human Nature [Part One of Two]
3.09 - The Family of Blood [Part Two of Two]
3.10 - Blink
3.11 - Utopia [Part One of Three]
3.12 - The Sound of Drums [Part Two of Three]
3.13 - Last of the Time Lords [Part Three of Three]
NaN - Children in Need: Time Crash [technically takes place during the events of 3.13]
The Complete Fourth Series - David Tennant as the Tenth Doctor
4.00 - Voyage of the Damned
4.01 - Partners in Crime
4.02 - The Fires of Pompeii
4.03 - Planet of the Ood
4.04 - The Sontaran Stratagem [Part One of Two]
4.05 - The Poison Sky [Part Two of Two]
4.06 - The Doctor's Daughter
4.07 - The Unicorn and the Wasp
4.08 - Silence in the Library [Part One of Two]
4.09 - Forest of the Dead [Part Two of Two]
4.10 - Midnight
4.11 - Turn Left [Part One of Three]
4.12 - The Stolen Earth [Part Two of Three]
4.13 - Journey's End [Part Three of Three]
The Complete Specials - David Tennant as the Tenth Doctor
S1 - The Next Doctor
S2 - Planet of the Dead
S3 - The Waters of Mars
S4 - The End of Time, Part One
S5 - The End of Time, Part Two
The Complete Fifth Series - Matt Smith as the Eleventh Doctor
5.01 - The Eleventh Hour
5.02 - The Beast Below
5.03 - Victory of the Daleks
5.04 - The Time of Angels [Part One of Two]
5.05 - Flesh and Stone [Part Two of Two]
5.06 - The Vampires of Venice
5.07 - Amy's Choice
5.08 - The Hungry Earth [Part One of Two]
5.09 - Cold Blood [Part Two of Two]
5.10 - Vincent and the Doctor
5.11 - The Lodger
5.12 - The Pandorica Opens [Part One of Two]
5.13 - The Big Bang [Part Two of Two]
The Complete Sixth Series - Matt Smith as the Eleventh Doctor
6.01 - The Impossible Astronaut [Part One of Two]
6.02 - Day of the Moon [Part Two of Two]
6.03 - The Curse of the Black Spot
6.04 - The Doctor's Wife
6.05 - Will post title 5/28 [Part One of Two]
6.06 - Will post title 6/4 [Part Two of Two]
6.07 - Will post title 6/11 [Part One of Two, SPRING FINALE]
6.08 - ??? [Part Two of Two, FALL PREMIERE]
6.09 - ???
6.10 - ???
6.11 - ???
6.12 - ???
6.13 - ??? [Season Finale]
"Shh! Eleventh!" Classic Doctor Who
Alright, here's what to watch when.
The Sarah Jane Adventures
Doctor Who - Season One (Christopher Eccleston and Billie Piper)
Doctor Who - Children in Need (Sometimes known as "Born Again" - David Tennant and Billie Piper)
Christmas Special 1 - The Christmas Invasion (David Tennant and Billie Piper)
Doctor Who - Season Two (David Tennant and Billie Piper)
Christmas Special 2 - The Runaway Bride (David Tennant and Catherine Tate)
Doctor Who - Season Three (David Tennant and Freema Agyeman)
Doctor Who - Time Crash (David Tennant and Peter Davison)
The first animated episode, "The Infinite Quest", takes place between 3.07, "42", and 3.08, "Human Nature".
Christmas Special 3 - Voyage of the Damned (David Tennant and Kylie Minogue)
The first season of Torchwood takes place roughly at the same time as the third season of Doctor Who. If you're watching it at the same time, try watching an episode of Torchwood after each Season 3 episode, starting with "The Runaway Bride". If you time it right, you should watch the Torchwood S1 finale, "End of Days", right before 3.11, "Utopia".
The first season of The Sarah Jane Adventures also takes place roughly at the same time as the third season. It doesn't cross over into the main Doctor Who plotline, however.
The Comic Relief special, "Time Crash", technically takes place during the season 3 finale. You can get away with watching it afterwards.
Doctor Who - Season Four (David Tennant and Catherine Tate)
The special Proms episode, "Music of the Spheres", takes place after the Season Four finale.
Christmas Special 4 - The Next Doctor (David Tennant and David Morrissey)
The second season of Torchwood takes place roughly at the same time as the fourth season of Doctor Who. Again, you should be done with the Torchwood season before the Doctor Who season finale starts in 4.11, "Turn Left".
The second season of The Sarah Jane Adventures takes place roughly at the same time as the fourth season of Doctor Who. It'd probably make the most sense to start watching after 4.05, "The Poison Sky".
The Torchwood miniseries, "Children of Earth", takes place at the same time as the following special episodes. If you want to know the story behind Jack's appearance in "The End of Time", watch "Children of Earth".
Similarly, season three of The Sarah Jane Adventures takes place at the same time as the specials, and features an appearance by the Doctor in "The Wedding of Sarah Jane Smith". If you alternate Sarah Jane and Doctor Who, you should reach "The Wedding of Sarah Jane Smith" before "The End of Time, Part 1".
Special 1 - Planet of the Dead (David Tennant and Michelle Ryan)
Special 2 - The Waters of Mars (David Tennant and Lindsay Duncan)
The second animated episode, "Dreamland", takes place between "The Waters of Mars" and "The End of Time, Part 1".
Special 3 - The End of Time, Parts 1 and 2 (David Tennant and Bernard Cribbins)
Doctor Who - Season 5 (Matt Smith and Karen Gillan)
The fourth season of The Sarah Jane Adventures takes place after the fifth season of Doctor Who, and features an appearance by the Doctor and Jo Grant in "Death of the Doctor".
Christmas Special 5 - A Christmas Carol (Matt Smith, Karen Gillan, and Arthur Darvill)
Doctor Who - Time and Space (Matt Smith, Karen Gillan, and Arthur Darvill)
Doctor Who - Season 6 (matt Smith, Karen Gillan, and Arthur Darvill)
There's a ton of material from the classic series, and naturally, varying opinions as to what to watch. Here are some tried-and-true classics for you.
William Hartnell as the First Doctor: An Unearthly Child, The Aztecs, The Time Meddler
Patrick Troughton as the Second Doctor: Tomb of the Cybermen, The Mind Robber, The War Games
Jon Pertwee as the Third Doctor: Spearhead from Space, Inferno, The Daemons
Tom Baker as the Fourth Doctor: City of Death, The Talons of Weng-Chiang, Logopolis
Peter Davison as the Fifth Doctor: Kinda, Earthshock, The Caves of Androzani
Colin Baker as the Sixth Doctor: Vengeance on Varos, Revelation of the Daleks, The Two Doctors
Sylvester McCoy as the Seventh Doctor: The Curse of Fenric, Ghost Light, Survival
If you want Paul McGann as the Eighth Doctor, you'll have to hunt down the movie, as that's all that he appeared in. (There are audio plays, etc., but that's the only time he played him "on film".)