I’ve not actually seen as much Classic Who as it might appear. Nonetheless, I’ll try to answer your question.
Firstly, understand that the vast majority of Classic Who is composed of stories that stand almost entirely alone from one another. Cross-story character development is minimal and stories almost never rely on the plots of earlier serials to make sense. Because of this, it’s easy (and recommended!) to jump between Doctors, TARDIS Teams, and production eras as you explore. Unless you’re embarking on some critical venture, watching it all in order is unnecessary and unwise.
Second, keep in mind the original conditions of transmission at the time the stories were broadcast. That doesn’t mean you have to take seven weeks to watch Doctor Who and the Silurians, but watching seven 25-minute episodes - nearly three hours - of the same sets, scripts, and plot elements back-to-back is going to burn you out. Spread longer stories out over a day or two. For shorter stories, like four-parters, break up the episodes throughout a day, slipping meals, walks, or other activities between the parts. There are a handful of stories that are compelling and well-paced enough to be watchable in “omnibus” format, but they’re absolutely the exception.
Third, don’t feel compelled to enjoy stories, companions, or other elements of the Classic Series based on “fandom consensus” or any other such rot. Doctor Who pre-2005 is a very different program from the Revival - in fact, it’s several different programs all through its run.
Lastly, while expanded-universe material is (almost) never necessary, it can often satisfy unexplored topics. The Big Finish Sixth Doctor audios in particular, including The Marian Conspiracy, The Holy Terror, and Jubilee, are far better than anything Colin Baker got to take part in on television.
With all that out of the way, here are some suggestions:
- The Ark in Space. You’d never believe how horrifying green bubble-wrap can be. Four 25-minute parts starring Tom Baker, Elizabeth Sladen, and Ian Marter
- Genesis of the Daleks is simply remarkable. At six 25-minute episodes with more than a few captures-and-escapes it can drag, but even in omnibus format it’s intensely watchable. Tom Baker, Elizabeth Sladen, and Ian Marter.
- The Ribos Operation. Sets up the following five stories of the “Key to Time” season/story arc, introduces Romana I fantastically, and is arguably Robert Holmes’ masterwork. Can drag a bit, though, if watched in a bloc. Four 25-minute episodes starring Tom Baker and Mary Tamm.
- City of Death is never a bad choice. It’s uproariously funny, wonderfully acted, and while the plot barely manages to cohere, you’ll be too caught up in the ride to notice. Four 25-minute parts, available on Netflix Instant in the US. Tom Baker and Lalla Ward.
- The Curse of Fenric. If you want to understand how Doctor Who went from stories like Pyramids of Mars to stories like Rose, here’s your not-so-missing link - it’s got more companion development in 100 minutes than Sarah Jane got in two-and-a-half seasons. The last story produced for the classic series (though not the last aired), it also features stellar guest performances and iconic setpieces you’ve surely seen slyly referenced all over Doctor Who. Benefits from some minor knowledge of Ace from Dragonfire, but you can just read a summary, really. Four captivating (and bingeable) 25-minute episodes starring Sylvester McCoy and Sophie Aldred.
That’s all I got right now. I’m certainly missing/skipping a lot - I’ve only recommended stories from two Doctors, after all - but these are fantastic places to start. Just don’t get hung up on “canon” and pace yourself, and you’ll have a splendid time!
All good suggestions. I’d add a few more, covering other Doctors:
The Time Meddler — one of the best, and funniest, Hartnell stories
The Mind Robber — a wonderfully clever metafiction which is to my mind the single best thing the programme ever did.
The Time Warrior — people usually mention this as the story where the Sontarans and Sarah Jane Smith make their first appearances and where Gallifrey is first named, but in fact it’s worth watching just because it’s funny, and clever, and one of the few truly great Pertwee stories.
The Five Doctors is a continuity-laden self-indulgent mess, so fans of the recent Moffat era might like it ;) More to the point, it’s fun, it’s meant to be watched in one sitting, and it’ll give a taste of what all the early Doctors are like.
And Vengeance On Varos, for the single best cliffhanger the TV series ever had.
Suggestions are more of a personal thing and everyone would have their own, but I’d just like to say that the first few paragraphs of this answer are just about the best summary of Classic Who to a newcomer that I’ve ever seen, and aid understanding far better than simply listing a load of stories and expecting them to be loved context-free by someone who has never watched a TV show that was produced prior to the 1990s.