There are two types of endgames, theoretical and practical and the way to study them is different.
I remember when I was a kid and was about to learn the theoretical endgames. The best manual at the time was Fine’s monumental work Basic Chess Endgames. I had it in two volumes, a 1951 edition in Croatian from Sahovska Naklada. Even though Averbakh’s tomes on endgames were also out by then, I didn’t have them so Fine it was.
My way was the pedestrian way. I simply played over all the examples in both books. I would set up the position and then I would play over the moves. I would pause and try to understand the principles that were explained in words. Then make the moves on the board, often repeating the process. After finishing the work (which lasted for what seemed an infinite amount of time) I noticed a marked improvement in my endgame play. As if some inner pieces of the puzzle fell in place and I was just playing better.
After some period I would repeat the process. It was a very long and not too interesting process, going through all the theoretical endgames, but I knew it was a useful one, so I did it. That is how I acquired my theoretical knowledge of endgames.
For the practical endgames, I am greatly indebted to Shereshevsky’s Endgame Strategy (Konturi Endshpilja in Russian). The book was full of general principles and good examples. Even though nowadays with the help of computers I discover many mistakes in the book, the main teachers were the players who played those endgames – Capablanca, Botvinnik, Smyslov, Fischer… By playing over their (end)games it seems that the subconscious picks up the invisible threads that are required to produce good endgame moves.
Apart from Shereshevsky, there were also other books and games of the great players. Three of them, in particular, were very impressive: Capablanca, Botvinnik, and Smyslov. Botvinnik’s three tomes (1923-1941, 1942-1956 and 1957-1970) were amazing as was Smyslov’s Letopis Shakhmatnogo Tvorchestva (Annals of Chess Creativity). (All of these are in Russian, as I studied them in the original). It was the recommended method back then – you study the (end)games of the great players and after a while, your general game improves.
Theoretical first, practical second
Nowadays there are new books. I usually recommend Jesus De la Villa’s 100 Endgames You Must Know. It is the most basic knowledge a chess player must have.
And then there is, of course, the Endgame Bible – Dvoretsky’s Endgame Manual. Dvoretsky is the ultimate guide to the theoretical endgames, it is what Fine and Averbakh used to be in pre-computer times, only this time the variations are error-free. Some time ago I did with Dvoretsky what I did with Fine many years ago, only this time it was tougher with all the computer lines involved. I have read that The Manual was Kasparov’s favorite book.
I would say that the order of studying endgames should be theoretical first, practical second. The rationale here is that you must know what you’re going for, what your end-position is, with a clear understanding and knowledge of its evaluation and method of play. These end-positions are the theoretical endgames and you must be able to rely on them without a shadow of a doubt. The actual memorization of these theoretical endgames is similar to learning the multiplication table. It is problematic at first, it requires effort, but they must be memorized and after that life is much easier! (Understanding the principles etc. goes without saying.) As with all things requiring memorization, repetition is something that you will need to do from time to time.
For practical endgames, the most important thing here is to develop a feeling for them. You get that feeling by playing over countless games by the great players. I’d recommend the players from the past (notably my favorites Capablanca, Botvinnik, and Smyslov) because their play is somehow easier to understand and assimilate. Today’s chess is just way too burdened by computer variations and is too complex.
I would like to finish with a modern method for getting better at endgames. It consists of trying to win a technically winning position against an engine. I remember I read somewhere that this was the method Topalov used before his stellar period in the mid-00s. I tried it myself and what I can say is that it is even more frustrating than solving Dvoretsky’s puzzles. In other words, I rarely, if ever, managed to win a game.
Now all that remains is to get down to work. Good luck!