There will be a thousand different views on this, and a million different ways of doing it. It is of course tempting to send you to the absolute highlights, the rarely disputed cream of the crop first. The issue with that though is that you then risk it being all downhill and have the bar set incredibly high. It can take some adjusting getting into the classic series, because of the serialised nature and the often (though not always) considerably slower pace. I like that you mention ‘each and every Doctor’ and I think this would be a good approach - maybe one serial for each Doctor to get a feel for each of them, and then go on from there.
There would be argument for starting with the TV movie (if you’ve not seen it already) since, though it’s not part of the original run of the show, it acts as a bridge (stylistically, not just temporally) between the vibe of the classic series and the vibe of the modern one.
I would then maybe move on to one of the colour-era Doctors rather than going back to the First, as he can be a little jarring against what you’re used to in the character. A nice, uncontroversial Fourth Doctor story maybe, since he is the general public’s favourite and it was a well-thought-of era. I’d suggest maybe The Ark in Space. That has the bonus of including Sarah Jane, so someone familiar you can carry over from what you’ve seen. Alternatively, The Robots of Death is a fun murder mystery in space!
Then you could go forward in Doctors from Four, and come back to the beginning once you’ve had a good flavour, as 4-7 are generally more palatable to a new viewer than 1-3 IMO. For the Fifth Doctor, there’s heaps of good choices available. Tian’s suggested The Five Doctors, which is a good shout - it was the first classic story I saw and it introduces you to five of the Doctors (sort of) and many companions. That said, there’s something to be said for leaving it until you’ve seen a lot more, so you can appreciate all the returning characters. It’s also one of the strongest episodes there’s been, which I’ve already mentioned is a risk. Alternatively, I’d suggest maybe Four to Doomsday, or, if you wanted something more of an opposite to The Ark in Space, the pseudo-historical, The Awakening.
The Sixth Doctor’s run was fairly brief and his second season is all one big story, so choices are limited. He’s also controversial, depending whom you ask (I love him and his era personally!) I would suggest Vengeance on Varos, or The Two Doctors, which would introduce you to the Second Doctor simultaneously.
Most of the Seventh Doctor serials would work as an intro as they’re fairly well contained individually. The Curse of Fenric is a popular one - his dynamic with Ace (his second companion) is famous and she is seen as something of a prototype for the modern companion. (On the other hand, Mel from the recent era was his first companion, so you may want to see what she was like back in the day - in which case, you could try Delta and the Bannermen), which gives you more of the wacky vibe of the early portion of Seven’s era.
Then back to the First Doctor. If you’ve got through the above and enjoyed them, then you’re ready for the black-and-white era (I love it personally, but it is very different). You could watch An Unearthly Child to see how it all started. If you’re using iPlayer however, that serial is absent, so you’d need something else. The Dalek Invasion of Earth gives a good flavour of both one of the most popular TARDIS teams of all time, and how the Daleks were handled back in their early days. It is longer than all my previous suggestions though, so settle in.
Many of the Second Doctor stories are at least partially missing and you’d definitely want to start with something that exists in its entirety. For something shorter and a bit nuts, try The Mind Robber. Or for something a bit more standard (but longer), maybe The Seeds of Death.
And then you reach the Third Doctor. By this point, if you’ve done one from every Doctor, you definitely deserve to watch something brilliant, so I would maybe go for Spearhead from Space. As Tian said, it’s the beginning of a whole new era of the show, something totally fresh and it holds up better than most (not least because it was all shot on film so the quality is higher). Then you’d be in a good position if you wanted to start watching things in order, to go through the Third Doctor’s era chronologically - though there are some long slogs in there, so you may prefer to keep jumping around. Alternatively, Terror of the Autons holds up well and is the first introduction of the Master…
There you go, there’s my personal recommendation. But others will hugely disagree, and they’re right to. As you say, there’s so much of it that it’s impossible to know where to start. Maybe the best thing to do is just choose one at random (you can use the Guide’s randomiser to do this) and see what you find! Of course there’s a risk it would start you off with something a bit sub-par, but you could always check what people think here beforehand - or after, if you didn’t like it and want to know if it’s representative of the whole era, or just a dud!
Sorry, a huge post. But there’s two suggestions for each Doctor. Maybe it’s some help to you, or to someone in the future. Who knows. I also don’t know how much you know about the classic show so if any of the above (‘serial’, ‘TV movie’, etc.) doesn’t make sense, just holler!
(I wouldn’t necessarily recommend the method, but it’s what suits me; I’ve been bouncing around finding some comics/stories released in the ‘60s as well to go along with.)