I recommend that all new players learn to keep score using what is called "algebraic" notation. In the diagram that follows, you will notice that the board includes the numbers 1-8 along the left edge and the letters "a" - "h" along the bottom the board. We can use the combination of one letter and one number to describe each square on the board.
For example, the white king begins the game on e1. The black rooks begin the game on a8 and h8. If I begin the game by pushing the pawn in front of the white king two squares, we can describe this move as e2-e4, or simply e4. Moving the knight on g1 to f3 would be written g1-f3, or more simply as Nf3.
An "x" is used to show that a capture has taken place. For example, Nxe4 means that the knight moves and captures whatever was on e4. You don't necessarily need to include "check" in your scoresheet, but you can easily with "ch" or just "+". For mate, many players use "#". Now who could resist that!