I know there are points for each chess piece, but is there a way to - imprecisely, for sure - measure/point chess moves?
Sure, that's how computer engines work.
For instance, if you start up ChessMaster and hit ctrl+m ("mentor"), it will give you a list of all the moves, and how the chess engine rates each of them.
How it works is that the engine looks as far as it can for every move, assuming best play for both sides, and rates the final position. That final position's rating is the value of the move.
To rate the position, the computer looks at material, space, pawn-structure, king-safety, known patterns and endgames, etc. etc. - basically, all the same things a human looks at, but the computer's not as good at it, because this is pattern matching, something computers are notoriously bad at. The only reason computers can beat humans is that the computer can see so far ahead so fast, and never makes mistakes.
Thus, looking a computer's rating of different moves might help settle an argument, or help point out mistakes in your games after-the-fact, but learning how computers rate positions is not going to help you get better.
Pawn = 1 point
Bishop and knight = 3 points
Rooks = 5 points
Queen = 9 points
But computers use a slightly more complicated way to count points.
Pawns, depending how many there are on the board (close or open position), depending on which file they are positioned, if they are passed pawns, if they are connected and on which rank they are, in endgame or opening, can have a relative value of 0,90 points to 3,5 points!
Bishops can have a better value than knights if the position is open and / or both bishops are on the board.
Queen and Rooks can have a slightly lower value if the position is closed while knights are worth more.
That's why some exchanges (knight for a bishop for example) are better for one or the other player.
Points (decimal points) will vary more or less depending on the chess engine.
I add points for movements of PIECES:
For instance, a rook is normally worth five points. But if has been moved to the "seventh" rank, (next to the opponents' pawns), I consider it a SIX point rook.
A "bishop pair," in an OPEN position against two knights or a bishop and a knight is worth more like SEVEN points, rather than six.
A knight is usually worth three points. But if it has found an outpost at say, e4, e5, or e6, maybe f5 or f6, in a "hole" from which it cannot be dislodged, it is a FOUR point knight.
A pawn is usually worth one point. But if has moved to a square past opponents' pawns so that it is a "passed" pawn, it is worth TWO points.
These are all significant advantages, enough to win a game BY THEMSELVES, even if everything ELSE is equal. Basically, you're an extra "point" ahead.