Set up a cutting area outside with some scrap pieces of wood. Never cut with a saw directly on top of the ground, and never cut while holding a piece of paneling in your off hand. The goal is to have three or four pieces of scrap wood set out along the ground so that you can set your pieces of panel on top of them once you are ready to cut, giving you a few inches of space between the wood and the ground, provided by the wood scraps.
Measure the paneling for wherever it needs to be placed on the stairs. For example, stair face pieces will be cut off straight according to the height of the risers, with a width that depends on the width of your stairs. Wall panels, on the other hand, will be angled cuts that follow the angle of the stair as it goes up and down along the wall.
Mark straight pieces with the square, pencil and tape measure. For angled cuts along the wall, mark the two sides of the angle (upper and lower where it angles with the staircase) and transfer those marks to the piece of paneling. Snap a chalk line between the two marks to define the angle and serve as a guide for cutting.
Set the gauge on the circular saw to the depth of the paneling you are cutting. Put on all of your safety gear and place your paneling on top of your scrap wood for the cutting area. Place one foot firmly on the paneling to hold it in place. Hold the saw firmly in both hands, turn it on and push it through the paneling along your marked lines.