Always start by marking the holes on which you get to take a stroke. Make a little dot somewhere within the box for the holes on which your course handicap will be used. (The "handicap" row of the scorecard tells you where to take strokes. If your course handicap is 2, then take a stroke on the holes marked 1 and 2. If it's 8, then on holes designated 1 through 8. More here). If marking the card in the manner of the top example, also divide each of those boxes with a slash.
Write down your strokes taken on each hole as you normally would. The gross score (your actual strokes played) goes on top. Then, on holes where you are taking a stroke, write your net score (your actual strokes minus any handicap strokes) below the gross score.
When you tally up the total, again write your gross score on top and net score below the gross.