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Use a base coat. Paint each stripe separately. Wait for the paint to thoroughly dry.
An easy way to paint in your stripes is to apply a base color to the entirety of the paper (retrace lines afterward, if necessary). Then, once your base coat has dried, you can paint different colored stripes over it.  To keep your lines clean when painting the stripes, try using painters tape on the borders.  Keep in mind that you should mark which stripe gets which color as it can get confusing. You can do this by dotting each line with the color it will take, or by marking them with bits of tape. This is a good method if you plan to paint each line a different color - or at least more than two. For example, if you want to alternate shades of blue, you could start with a light blue at the top and then paint each stripe a darker blue until the bottom. This would give you an ombré look. Make sure your lines are clearly marked so your colors don't run over and into each other. Depending on the thickness of your paint, this can take anywhere from 10 minutes to a half hour. Once it has dried, gently pull up the tape if you used any. You can also take the time to erase any remaining pencil marks from your original grid.