I'm sure Fedya would love to [learn the basics], and so would many of us but nobody is willing to say what we should know inside and out in order for us to take your advice.
I think it's been said many times. Assuming we are talking about players who are well beyond the pure beginners' stage, it's making a big niche in whatever time you do use for practise for playing over pro games.
However, I do get the impression that many people seem to think this means serenely sitting before a kaya board methodically placing clamshells and slate according to a butterfly-bound edition of Go Seigen's collected games, all in the hope that by some magical osmosis the fragrance of great go will seep into the brain.
It's actually meant to be hard work with lots of reference to other books or other people. In some eays it's like writing an essay at college. You can crib whole paragraphs from the internet and might even absorb a titbit or two, or you can go the library, consult books, think about them, consult some more, write the essay, leave it then rewrite it. I need hardly say which method is more likely to help you learn the subject.
But in other ways, learning go is like learning a language. You can learn whole lists of vocabulary, much of which you may never use, or you can go to the country concerned, speak the language every day, and learn
only the words you actually use and need. By definition you are learning what is basic for you.
If you study go, you can go to a classic tsumego collection and work hard at it. Not a total waste of time, but not very efficient. You'll learn under-the-stones patterns that you may never see in real life. The carpenter's square or a variation of it comes up very often (I did count once and I seem to recall it was about 4% of the time). Under-the-stones comes up close to zero times. In my experience most dan players can solve under-the-stones problems but almost none know the ins-and-outs of the carpenter's square (my self included

)
But if you play over games and come to, say, a carpenter's square, and you are not sure about it, you look it up in your library, you maybe examine the variations on another board. You are still studying tsumego but, best of all, you see the position in the context of a real game and so don't learn just techniques but e.g. timing of life & death manoeuvres, their relation to other parts of the board (e.g. the different ways to connect the bigger L groups), comparative sizes, best plays to leave aji and ko threats. These are all basics because these are the things that come up most often, and you are seeing them in the precise frequency that they occur in real games, so your learning is as efficient as can be.
You can apply the same thinking to tesujis, fuseki, the endgame and everything else.
I'd suggest that the best question to put to other forums is what they recommend as
reference works. Not books to read (although they have their place) but books that are well structured and well indexed, books that are comprehensive. I'm not entirely au fait with those in English but I think you need a joseki dictionary, a fuseki dictionary and Fujisawa's tesuji books to start with, and many other items you can find easily on Sensei's Library (e.g. you see a life & death position with a notch in a game - turn to notchers on SL and look it up - then go back to hard study of the game where you can reflect on how this position arose (from a systemic weakness?) and what effect it will have on the rest of the game. You can even look up things like players' styles on SL.
In short, I'm saying that it is useful to regard whatever comes up most often as the most basic.