How can one easily learn chess board coordinates?

While specialized training may be useful, you also need to be in a situation where that training can be used.

I'm mainly interested in chess compositions, and had much the same problem following solutions, or setting up problems from notation (i.e. not from diagrams, but from textual description such as "White: K at KR3, Q at KB2, ..." and so on. Descriptive or algebraic notation.)

I found that while I could get the file right pretty easily, the rank required counting. But after a while, it became obvious that ranks 6, 7 and 8 were easier to count backwards than forwards, (i.e. as 0, 1, 2 from the back rank) and doing that adjustment mentally was quite easy. Getting ranks 4 and 5 right was a bit more of a problem, and I mixed them up for the longest time. But persistence pays off. I think it took me about a year of more or less daily use to make errors reasonably rare. (Now I mix up black and white coordinates in descriptive notation instead ...)

But that's me. I suspect it's fairly individual -- I started more or less from zero.

Playing blindfold chess might also help with coordinates a lot, once we have a visual sense of where the squares should be, and how they are related spatially in the mind, we can get really comfortable using the chess algebraic notations.

If you just want to learn notation, its a-h horizontally, and 1-8 vertically on the 8x8 board.

Memorize not just squares, but ranks and files. a: Queen's Rook b: Queen's kNight, c: Queen's Bishop, d: Queen, e: King, f: King's Bishop, g: King's kNight, h: King's Rook. 1: White's back rank, 2: White's pawn rank, 3: White one pawn-move, 4: White's two pawn-move. For each square, you should become familiar what role that square in particular play, but also what role its rank and file plays. For instance, since Black usually castles into g8, f6 the square a knight can attack the king, and where a pawn can attack the middle of the three f7, g7, h7 pawns, etc. It's where a bishop can threaten mate when combined with a queen on the g file, etc. The f file is just off the center on the king side. It's where you're likely supporting you e pawn from, and losing control of it means that attack can come into the g file (where you king usually is). Rank 6 is the last rank before Black's pawns.

Start with the edges of the board: a file, h file, rank 1, rank 8. It should be easy to get a1, a8, h1, and h8 down. Then go for the middle: rank 4, rank 5, d file, e file. Memorize d1, e1, d8, e8, then d4, d5, e4, e5, then move on to a4, a5, h4, h5. Every other square is only one King's move away from one of these 16 squares.

I see from your profile that you are familiar with programming concepts. As a programmer myself, familiar with hexadecimal and other number base notations, I have found it easier to convert the alphanumeric chess position to two numbers first before locating the position.

So when you see an a, b, c, d, e, f, h convert it to a number in your head first 1, 2, 3, 4, 5, 6, 7, 8.

Then you look for square [3, 4] or [7, 1] remembering that the column is first, then the row.

So all you need to learn is to quickly convert the following:

a=1
b=2
c=3
d=4
e=5
f=6
g=7
h=8

So when you see a position like c3, immediately say [3, 3] in your head before looking for it.

This was especially easier for me when I'm playing the black pieces and need to count from the top right corner, as it is quicker than trying to count alphabet letters from right to left. But like everyone else said - it takes practise before it becomes natural.

Some websites offer "coordinate drills". I'm not sure how effective they are for learning, but you can certainly give them a try. I found it to be surprisingly entertaining to try to beat my personal record number of coordinates per minute (or whatever the time limit is). Here are the two I know of, but maybe there are others (and maybe there are apps, as well):

https://www.chess.com/vision

https://lichess.org/training/coordinate

Personally I've found that studying openings (and to a certain extent, checkmate patterns) helps learn coordinates because you start associating coordinates with certain moves, which gives each square its own "personality". For example, c4 is not just c4 anymore; it is the square where the bishop goes in the Italian game, or where the pawn goes in the queen's gambit.

This is something that you gain through experience. One good way to obtain that kind of experience would be to buy a large game collection - for example, a modern algebraic edition of the 1953 Zurich tournament book by Bronstein - and play through them. Not only will the experience accustom you to finding each square, you'll undoubtedly learn something from playing through so many master games.

This might disappoint you but the only thing you can do in the beginning when learning coordinates is to count, since you will lack the intuition necessary to identify coordinates immediately. As you gain chess experience the intuition necessary to quickly determine the coordinates of a square will be built up to the point where you know exactly where (for instance) f6 is on the chessboard without having to think about it.

But in the beginning (as with anything else) you will have to spend some mental effort to locate squares on the chessboard by their coordinates. To make the learning process quicker you could try to incorporate identifying the coordinates of squares while playing (for instance by keeping a scoresheet of the games you play) but other than that there isn't really a quick method to just learn this without effort.