A move here shall be defined as the FIDE super-long notation of a legal chess move: Sg1-f3, O-O, e5xd6 ep, Da8xg8+, Rd1-d6# and whatnot. I.e. starting square, ending square, capture sign, check sign, mate sign, ep (all if applies), castling. (Stalemate is usually not notated.) How many moves are possible? Danger, counting pitfalls: Bh1-a8 is never a check!
(I think this question might have been addressed in a problem chess journal already.)
OK, let's break it down piece by piece, first calculating without check/checkmate and then with.
Pawns
Each pawn has 6 non-capturing non-promoting moves, and can promote to 4 different pieces. The rook pawns have, including en passant, 6 non-promotion captures and 4 promotion captures. The non-rook pawns have double the captures. This gives us 20 moves per rook pawn and 30 per non-rook pawn. There are 4 rook pawns and 12 non-rook pawns. 4 times 20 is 80, 12 times 30 is 360, added together is 440.
Every pawn move, including en passant and promotions, could be a check or checkmate. So triple that. 440 times 3 equals 1320 total pawn moves.
Knights
There are 16 squares a knight can move from which give it 8 places to go, 16 squares where it has 6 places to go, 20 squares where it has 4 places to go, 8 squares where it has 3 places to go, and 4 squares where it has 2 places to go. So that's 128 + 96 + 80 + 24 + 8 = 336.
There is no particular reason why a knight move couldn't be check or checkmate, and no particular reason why it couldn't be a capture, so multiply this by 6. 336 times 6 is 2016 total knight moves.
Bishops
From the center 4 squares, a bishop has 13 moves. From the 12 squares ringing the center, a bishop has 11 moves. From the 20 squares ringing those squares, a bishop has 9 moves, and from the 28 squares at the edge of the board, a bishop has 7 moves. So that's 52 + 132 + 180 + 196 = 560 bishop moves.
As warned in the question, some bishop moves cannot be check or checkmate. These would be the moves where the bishop cannot uncover another piece and does not control any new squares itself. Those moves turn out to be only the moves from one corner to another, of which there are 4. So, subtract those 4 before multiplying, 556 times 3 is 1668, add back those 4, and get 1672. Any of those could also be a capture, so 1672 times 2 = 3344 total bishop moves.
Rooks
Rooks are easy. Each rook has 14 moves from any spot on the board. 14 times 64 is 896.
Any of these moves could be a check or checkmate, and any of those could also be a capture, so 896 times 6 is 5376 total rook moves.
Queens
Queens can do any move the rook or bishop can do. 560 bishop moves plus 896 rook moves equals 1456.
Queens can give check on those corner to corner moves where the bishop can't, so all of these moves could be check or checkmate. And any of those moves could also be a capture. So we can simply multiply this by 6 and get 8736 total queen moves.
Kings
There are 36 squares where a king has 8 possible moves, 24 edge squares where it has 5 possible moves, and 4 corner squares where it has 3 possible moves. There are also 2 castling moves, which are classified as king moves. (I assume White castling and Black castling count as the same move since they use the same notation, even though the pieces move to different squares.) 288 + 120 + 12 + 2 = 422.
Kings can never give check directly; they can only give check if uncovering another piece - or by castling, where the rook can give check. The only way they can't uncover a piece is if they move from the corner (12 moves) or non-diagonally from an edge square to an edge or corner (48 moves). So subtract 60, 362 times 3 is 1086, add the 60 back, to get 1146. All of these except castling could also be captures. So there are 2286 total king moves.
Sanity check
Let's make sure these numbers make sense in relation to each other.
There are more queen moves than anything else, rook moves are second, bishops are third, kings are fourth, knights are fifth, and pawns are last, which makes intuitive sense.
Total
Now we can add it all up. 1320 + 2016 + 3344 + 5376 + 8736 + 2286 = 23,078 total moves.
