The Fast Track
For when you want a guided tour that covers a smaller amount of stuff.
This is Incomplete! Work In Progress!
Segment 1: 
Monthly Adventures: An Introduction 
(1999-2000)
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The Monthly Adventures is a series of standalone Doctor Who stories. Each story consists of four half-hour parts, and features one of the classic doctors. In general, you can choose any story and just listen to it – only a few of them are designed to lead into each other.
The very first one is far from the best thing they’ve done, but it’s a fun starting point:
The Sirens of Time
Featuring: the 5th, 6th, and 7th Doctors
Standalone: Yes
Here are some of the first few, that I consider worth listening to:
Whispers of Terror
Featuring: the 6th Doctor, Peri
Standalone: Yes

The Fearmonger
Featuring: the 7th Doctor, AceStandalone: Yes
The next audio drama introduces a unique-to-audio companion, Evelyn:
 The Marian ConspiracyFeaturing: the 6th Doctor, Evelyn
The Marian ConspiracyFeaturing: the 6th Doctor, EvelynStandalone: Yes
Essential
 (plotline: Evelyn)
And then some assorted episodes:
The Genocide Machine
Featuring: the 7th Doctor, Ace, Bev, Daleks
Standalone: Yes
Essential
 (plotline: Dalek Empire)
The Spectre of Lanyon Moor
Featuring: the 6th Doctor, Evelyn, Brigadier
Standalone: Yes
Note: This drama includes graphic and potentially upsetting scenes of heavy violence.
The Apocalypse Element
Featuring: the 6th Doctor, Evelyn, Romana, Time Lords, Daleks
Standalone: Yes
Essential
 (plotlines: Dalek Empire, Gallifrey)

The Holy Terror
Featuring: the 6th Doctor, FrobisherStandalone: Yes
 The Mutant PhaseFeaturing: the 5th Doctor, Nyssa, Thals, Daleks
The Mutant PhaseFeaturing: the 5th Doctor, Nyssa, Thals, DaleksStandalone: Yes
Essential
 (plotline: Dalek Empire)
Bonus
Some segments will have a bonus section. These are audio dramas that belong in this segment, but can be skipped if you’re trying to go super fast.
Red Dawn
Featuring: the 5th Doctor, Peri, Ice Warriors
Standalone: Yes
The Fires of Vulcan
Featuring: the 7th Doctor, Mel
Standalone: Yes
The Shadow of the Scourge
Featuring: the 7th Doctor, Ace, Benny
Standalone: Yes
Segment 2: 
The Eighth Doctor joins 
(2001)
The Eighth Doctor joins the audio adventures, with new companion Charlotte Pollard (Charley)!
Unfortunately, the first “season” of adventures is a bit hit and miss. The first episode is good:
Storm Warning
Featuring: the 8th Doctor, Charley
Standalone: Yes
Essential
 (plotline: Charley)
But the episodes after that get mixed reviews.
Bonus
I enjoyed the second episode, but not the third. Other people have the opposite opinion. Feel free to either listen to these or skip them:
Sword of Orion
Featuring: the 8th Doctor, Charley, Cybermen
Standalone: Yes
The Stones of Venice
Featuring: the 8th Doctor, Charley
Standalone: Yes
The fourth episode, Minuet in Hell is generally considered to be not good at all.
Segment 3: 
More Monthly Adventures 
(2001)
Here’s some more adventures with doctors 5-7.
 Dust BreedingFeaturing: the 7th Doctor, Ace, Bev, the Master
Dust BreedingFeaturing: the 7th Doctor, Ace, Bev, the MasterStandalone: Yes
Essential
 (plotline: the Master)
Project: Twilight
Featuring: the 6th Doctor, Evelyn, Nimrod, the Forge
Standalone: Yes
Essential
 (plotlines: Evelyn, Forge)
Note: This drama includes graphic and potentially upsetting depictions of heavy violence
 The Eye of the ScorpionFeaturing: the 5th Doctor, Peri, Erimem
The Eye of the ScorpionFeaturing: the 5th Doctor, Peri, ErimemStandalone: Yes
Essential
 (plotline: Erimem)
Note: This drama includes graphic and potentially upsetting depictions of torture
 ColditzFeaturing: the 7th Doctor, Ace, Klein
ColditzFeaturing: the 7th Doctor, Ace, KleinStandalone: Yes
Essential
 (plotlines: Ace, Klein)
Note: This drama includes graphic and potentially upsetting depictions of heavy violence, in addition to implications of sexual assault
Bonus
The One Doctor
Featuring: the 6th Doctor, Mel
Standalone: Yes
Segment 4: 
Dalek Empire I 
(2001)
The Dalek Empire series asks the question “What if the Daleks invaded, and the Doctor wasn’t there to stop them?”.
The Dalek Empire stories were set up in a thread through 4 Monthly Adventures (The Genocide Machine, The Apocalypse Element, The Mutant Phase, and The Time of the Daleks), but the series itself can be listened to without that prior context.
Warning: These stories feature a lot of death. Also, since they form a continuous story (unlike most other releases, which are standalone) you need to listen to them in order.
 Invasion of the Daleks
Invasion of the DaleksStandalone: Yes
Essential
 The Human Factor
The Human FactorStandalone: No
Essential
 "Death to the Daleks!"
"Death to the Daleks!"Standalone: No
Essential
 Project Infinity
Project InfinityStandalone: No
Essential
Segment 5: 
Anti-time 
(2002)
Some more stories, featuring the 8th Doctor and Charley:
 The Chimes of MidnightFeaturing: the 8th Doctor, Charley
The Chimes of MidnightFeaturing: the 8th Doctor, CharleyStandalone: No
(first listen to: Storm Warning)
Essential
 (plotline: Anti-time)
Note: This drama includes graphic and potentially upsetting depictions of death, suicide, and self-harm
 Seasons of FearFeaturing: the 8th Doctor, Charley
Seasons of FearFeaturing: the 8th Doctor, CharleyStandalone: No
(listen to the Anti-time plotline in order)
Essential
 (plotline: Anti-time)
 Embrace the DarknessFeaturing: the 8th Doctor, Charley
Embrace the DarknessFeaturing: the 8th Doctor, CharleyStandalone: Yes
Essential
 (plotline: Anti-time)
Note: This drama includes graphic and potentially upsetting body horror involving eyes
 The Time of the DaleksFeaturing: the 8th Doctor, Charley, Daleks
The Time of the DaleksFeaturing: the 8th Doctor, Charley, DaleksStandalone: No
(listen to the Anti-time plotline in order)
Essential
 (plotlines: Anti-time, Dalek Empire)
Neverland
Featuring: the 8th Doctor, Charley, Romana, Time Lords
Standalone: No
(listen to the Anti-time plotline in order)
Essential
 (plotline: Anti-time)
Bonus
Living Legend
Featuring: the 8th Doctor, Charley
Standalone: Yes
Segment 6: 
More Monthlies 
(2002-2003)
Some more monthlies:

Spare Parts
Featuring: the 5th Doctor, Nyssa, CybermenStandalone: Yes
Note: This drama (like most cybermen stories) includes graphic and potentially upsetting depictions of body horror

...ish
Featuring: the 6th Doctor, PeriStandalone: Yes
Note: This drama contains a graphic and potentially upsetting depiction of suicide
The Rapture
Featuring: the 7th Doctor, Ace
Standalone: No
(first listen to: Colditz)

The Sandman
Featuring: the 6th Doctor, Evelyn, the GalyariStandalone: Yes
Note: This drama contains a potentially upsetting use of a slur against Romani people
The Church and the Crown
Featuring: the 5th Doctor, Peri, Erimem
Standalone: Yes
Note: While it is standalone, it is better to listen to "The Eye of the Scorpion" first. This drama includes a graphic and potentially upsetting depiction of torture

Jubilee
Featuring: the 6th Doctor, Evelyn, DaleksStandalone: Yes
Note: This drama includes graphic and potentially upsetting depictions of sexism, abusive relationships, heavy violence, and body horror

Doctor Who and the Pirates
Featuring: the 6th Doctor, EvelynStandalone: Yes
Note: This drama seriously discusses mental health, depression, self-harm and suicide.
Project: Lazarus
Featuring: the 6th and 7th Doctors, Evelyn, Nimrod, the Forge
Standalone: No
(first listen to: Project: Twilight)
Essential
 (plotlines: Evelyn, Forge)
Note: This drama includes graphic and potentially upsetting depictions of heavy violence and torture
Flip-Flop
Featuring: the 7th Doctor, Mel
Standalone: Yes
Note: While this is one of my favourite time travel/paradox stories, be aware that this story includes antisemitism
Bonus
The Dark Flame
Featuring: the 7th Doctor, Ace, Bernice, Joseph
Standalone: Yes
Creatures of Beauty
Featuring: the 5th Doctor, Nyssa
Standalone: Yes
Segment 7: 
Dalek Empire II: Dalek War 
(2003)
The Dalek Empire series continues.
 Dalek War: Chapter One
Dalek War: Chapter OneStandalone: No
Essential
 Dalek War: Chapter Two
Dalek War: Chapter TwoStandalone: No
Essential
Note: This drama contains potentially upsetting depictions of torture and suicide
 Dalek War: Chapter Three
Dalek War: Chapter ThreeStandalone: No
Essential
 Dalek War: Chapter Four
Dalek War: Chapter FourStandalone: No
Essential
Note: This drama contains potentially upsetting depictions of suicide
Segment 8: 
The Villains 
(2003)
Some episodes focusing on villains:
Omega
Featuring: the 5th Doctor, Omega
Standalone: Yes
Davros
Featuring: the 6th Doctor, Davros
Standalone: Yes
Master
Featuring: the 7th Doctor, the Master
Standalone: Yes
Zagreus
Featuring: the 5th, 6th, 7th, and 8th Doctors
Standalone: No
(first listen to: Neverland)
Essential
 (plotline: Anti-time)
Everything from this point on is guesswork.