Only Seasons 1-6 will be covered. Season 7 and on exist fully.
Trying to avoid major spoilers here.
Season 1
An Unearthly Child/10000 BC- Watch the first episode, definitely. The other three episodes are skippable, but they are extant.
The Daleks- If you skip this one, slap yourself stupid.
Edge of Destruction- Not much happens, but it's short, and introduces plot points important later on. Decent character development too.
Marco Polo- Completely Missing. Which is a damn shame, since it's one of the funnier Historicals. Plods in the middle, but I, at least, enjoyed it.
Keys of Marinus- Very odd serial, it's closer in format to the format of the revival series, of several episodic quests linking to a larger one. Can be skipped without affecting the understanding of the characters and plots, but it's fun.
The Aztecs- Utterly Boring. Not missing, but utterly boring, which is a shame, because it's largely Barbara centric. Has a Doctor romance subplot thing going. Don't think anything brought up in this series applies anywhere else.
The sensorites- Has the first clues that the Doctor and Susan are other than human.The first descriptions of what would later become Gallifrey. I will forever hate the Sensorites because I'm pretty sure they made the Ood (who make-a no damn sense and hail from the same system as said Sensorites) but other than that, not bad for 1st season.
The Reign of Terror- The Doctor and Co in Revolutionary France. Keep Susan's complaints about the textbook in the first episode in mind here.
Season 2
Planet of Giants- Tiny Tardis. Present Day Earth. Amusing for size shennanigans and parallel stories, but skippable.
The Dalek Invastion of Earth- Exactly what it says on the tin. Occurs in 2150. You will watch this. First cast departure.
Afterwards, you will watch the two Peter Cushing movies and laugh, because they are silly.
The Rescue- New Cast, Yay! Short, but enjoyable.
The Romans- A historical in which the Tardis gang decides to get some down time in 1st century Rome. They get into trouble as a result, of course. Deliberately farcical, in places, contrasting with some dark elements, in an odd effect.
The Web Planet- What is this I don't even. A six episode serial about a planet filled with sapient talking giant insects. The most honest thing I can say is it is weird. Does not affect the overall plot much at all. Entirely skippable.
The Crusade- Doctor Who meets Assassin's Creed. More seriously, it's the 3rd crusade. The gang has to deal with with King Richard and Saladin and their entourages.
Space Museum- Another somewhat silly romp episode, the Doctor and crew find themselves in a space museum, experiencing people who they can see but cannot see them, until they find themselves and the TADIS as exhibits in the museum.
The Chase- Involves Daleks. Is partly missing. You can skip the christmas episode. First appearance of a future companion... playing another part entirely. Also marks the departure of Ian and Barbara and the taking on of a new companion. Definitely catch.
The Time Meddler- The Doctor meets one of his own people who is toying with history in decidedly unsubtle ways in 1066. Do catch. I love the meddling Monk.
Season 3
Galaxy 4- Entirely missing. Entirely skippable.
Mission to the Unknown- A one off episode. Features NONE of the regular cast. Is still very important.
The MythMakers- Entirely missing. Despite that, I still encourage watching. Covers the end of the Trojan War. Companion Exit.
The Dalek's Master Plan- Hi, I'm a story with Dalek in the title! You're going to watch me anyways!
The Massacre of St. Bartholomew's eve- France 1572. The Protestant Purges. Contains an absurd mistaken identity subplot, with the actual doctor missing for most of the series. A new companion arrives at the end, which is worth watching. The rest is Meh.
The Ark- Mankind is now travelling from earth to new worlds around new stars for mass colonization due to their expanding sun (This is probably them just being wrong, since both regular physics and the rest of the series contradicts this with an alternate reason for migrations out at this time). But trouble is afoot in the form of his companion's cold virus. and the doctor has to stop it... then they come back 700 years later and deal with new trouble.
The Celestial Toymaker- Weird weird weird as hell and loving it. Largely missing the Doctor except in voice (And his hand, played by someone else), an evil megapotent jackass wants to kill the doctor and companions through a series of rigged games. Almost entirely missing. The final episode exists.
The Gunfighters- Doctor Who in the old west. The Earps, Doc Holliday, etc. Not bad, but nothing special.
The Savages- It's the future, and a tech centric society and wild society are at each other's throats. Pretty missable. Exit of a companion, in case you skip.
The War Machines- Watch this one. Remember it. Be amused much later on. Introduces two new companions. Doctor's other old companion exits. Thinking computer gets an ego, wants to take over the world. Doctor says no.
Season 4
The Smugglers- Entirely Missing. Fairly Missable. Establishes new companions Ben and Polly a bit, but nothing vital.
The Tenth Planet- WATCH THIS. Cyberman appear here first, and the first regeneration occurs at the end of this series. Last episode is missing apart from the regeneration.
The power of the Daleks- Why, It's a new doctor and an old foe. Watch it. It's missing in its entirety. Watch it.
The Highlanders- Meet Jamie. The Second Doctor's companion. The last pure historical in the series until the 5th doctor's 2 part murder mystery. It's fragmented, but it introduces the Jamie, who, as I mentioned, IS the 2nd Doctor's definitive companion.
The Underwater Menace- Atlantis is rising! Only ep 3 is intact. Fairly weak, all told. Sad that of the 2nd doctor, thus far, the one to skip is the one that's most intact.
The Moonbase- A-ha! A partly extant 2nd Doctor Story! And one I reccomend. Remember those ugly as Cybermen from the 10th planet? They're back and a bit better blinged out with a plan to conquer the moon.
The Macra Terror- Remember the giant enemy crabs from Gridlock? This is their only other appearance. Mostly incomplete. Moving on.
The faceless ones- An identity theft plot involving Aliens at a London airport. Missing 3 episodes. Worth watching, however. Ben and Polly depart.
Evil of the Daleks- Daleks! Comedy! New companion Victoria! Only one episode remains! Despite that, quite watchable.
Season 5
Tomb of the Cybermen- Yay, more shiny tin men! This all exists. This also introduced Cybercontrollers and Cybermats, ganking folks and shinying them up. Also, the first explicit mention to the doctor's age- "about 450." IIRC, the 4th doctor claims to be younger than this before being called on it.
The Abominable Snowmen- A fun romp. All but ep 2 is lost. Introduces the Yeti, a recurring (though not often recurring) enemy that is important later.
The Ice Warriors- Introduces the ice warriors of mars. Is actually pretty dull in the middle, so episodes 2 and 3, which are missing, can be glossed over.
The Enemy of the World- The 2nd Doctor must play James bond against a man who looks exactly like himself, named Salamander. No. Really. 5 episodes missing. Onto the next serial, unless you really want to watch this one.
The Web of Fear- Why HALO THAR Colonel Lethbridge-Stewart. Now that's a young military man GOING places! Watch this one. It's pretty good, despite missing 5 of 6 episodes.
The Fury From the Deep- Why HALO THAR rather unimpressive Sonic Screwdriver actually used to unscrew a screw. Victoria Exits. Watch the beginning and the end if you want to skip this.
The Wheel in Space- New companion, Zoe. Cybermen! Space Station! Mostly Missing, like almost all of Troughton's era! Still worth watching a well reconstructed version.
Season 6
The Dominators- The Doctor and company come to the planet Dulkis for a holiday when they are interrupted by the murderous forces of the Dominators and their murderbots the Quarks, who want to conquer Dulcis. A complete standalone in terms of continuity, nevertheless, it's a good set of five completely extant episodes. The 'mental images' quip at the beginning is an in universe explanation for a repeat of 'The evil of the daleks' having been shown after Wheel in Space and before this serial.
The Mind Robber- This serial is a head trip. After the previous serial, a mechanical fault leads the Doctor to Land the Tardis outside of reality for safety. Things get weird from here as they land in the land of fiction. Includes several episodes of Jamie played by a different actor (with an in universe explanation for such) while he had CHicken Pox. The Villian is called The Master. Is not the The Master. I like this one. It's crazy. It's also fully intact.
The Invasion- WATCH THIS SERIES. 2 episodes are missing, the 1st and 4th. If you can get the animated versions of these eps, so much the better. First appearance of UNIT.
The Krotons- It exists in its entirety. Features machine overlords forcing a planets best and brightest into their servitude.
The Seeds of Death- More Ice Warriors! Much better than their previous appearance. Exists in its entirety.
The space Pirates- All but one episode missing. I wouldn't bother.
The War Games- Last 2nd Doctor Serial. Bother. Bother A LOT. Plus, it all exists! In short, the doctor plops down in what is apparently WWI, except it's very obviously the wrong time and planet for WWI to be happening. We see a little bit of Gallifrey at the end, though it's not called that yet.
