This is in part because I'm a believer in context, but also because while the new series gives you enough to go on, for the most part it doesn't fill in the backstory very much on a lot of topics because it doesn't have the time. (The Cybermen being an exception -- their alt-universe backstory reboot is actually pretty good.)
But this got me to thinking. Say you have someone who has enough general idea of what DW is about simply by being exposed to fans for a time, and you want to introduce them to a particular pillar of the show. Which stories would you pick? Never mind whether the story is considered one of the series greats or not, it just needs to have a reasonable entry point for a newbie. Mostly I'm thinking about the major villains, but also things like the Time Lords and regeneration.
The obvious answer here is to start with "The Daleks", but I reject it. For one, a lot of the early Hartnell era is an acquired taste to begin with. And for two, the Daleks came to be portrayed rather differently as the series matured. If it still existed, I think I'd say "The Power of the Daleks" because it builds up so well to showing the characteristics that define them. "Resurrection of the Daleks", "Revelation of the Daleks", and "Remembrance of the Daleks" are both fantastic, but bad for introductions since you need to know about Davros to understand why there are two Dalek factions. Jon Pertwee and Terrance Dicks never really believed in the Daleks and it shows. So that leaves the superlative "Genesis of the Daleks". Even though the sort of prologue at the beginning is easier to understand if you know who the Daleks are, once you're past that you get a good buildup of what they're about and you meet Davros at his 'best' as well.
This one was tougher than I thought. Obviously "The Tenth Planet" is out -- unless the BBC gives us an animated restoration of Episode 4 in the near future. And that's kind of sad because it would also be a good candidate as a regeneration story. There's "Tomb of the Cybermen" which is good (and a great showcase for Pat Troughton's Doctor to boot). "The Invasion" could work, too. There's a part of me that wants to use "Earthshock" since it's a good reintroduction, but the whole Adric thing would confuse a new viewer. "Revenge of the Cybermen" and "Silver Nemesis" are both crap... And "Attack of the Cybermen" works best if you already have seen the B&W stuff. I guess I will settle on "Tomb of the Cybermen" after all, because it fills in the backstory better than "The Invasion".
I really feel like this is a case where you begin at the beginning and "Terror of the Autons" is the place to start. I can make a case for better 'Master' stories but Roger Delgado was the first and the best, and the interplay between him and the Doctor is magnificent here. "The Deadly Assassin" is fantastic, but it really isn't the character's finest hour. And "Logopolis" is far, far too chock full of other things going on to start someone new on for this purpose. I'm sure that someone reading this will be utterly baffled that I didn't pick "The Daemons", but just don't see that as a great place to start.
The Time Lords
Although I keep wanting to come back to "The War Games" since it introduces us to so many things about them, I have to go with "The Deadly Assassin" as it really marks the start of the 'modern' interpretation of Gallifrey which has been built upon ever since. Which makes for an interesting recursion: How do you introduce a story to a newbie about the Time Lords when you sort of need to know about the Master already, and how do you show a newbie a story about the Master when you sort of need to know that he's a Time Lord?
This one's a bit tricky, because ideally you want a story that explains regeneration, not necessarily a story where a regeneration takes place. There's an urge here to go with the whole "The Keeper of Traken"/"Logopolis"/"Castrovalva" arc except this would make our excerpts very Master-heavy. It'd also make for a marathon of 12 episodes. And as great as "The Caves of Androzani" is and despite it having what I think is the best regeneration of all of them, like much of '80s DW it does sort of act like you already know the concept. "Spearhead from Space" would be a good candidate, but I discount it since there isn't an actual regeneration scene. So despite the atrocious alien-world sets I think I'll settle on "Planet of the Spiders".
Ah, UNIT. Here's another one that begs a knee-jerk nod to "The Daemons". And it does showcase the so-called 'UNIT Family' (The Brig, Sgt. Benton, Capt. Yates, Jo, & the Doc vs. the Master) but it really isn't a very good story for UNIT as an organization as they're pretty ineffective throughout. Although, if I ask the question 'When was UNIT effective?' does not bring a story immediately to mind -- with the exception of "The Invasion". But since UNIT is really a Pertwee-era thing, I would reject that. "Inferno" perhaps? "Spearhead from Space"? How about "The Mind of Evil"? Or maybe "The Daemons" after all. Huh.
So that would be 0 Hartnells, 1 Troughtons, 3 Pertwees, 2 Tom Bakers, and 0 Davisons, Colin Bakers, and McCoys. Interesting, that.