Remove the old carpeting from the stairs, if necessary. Using your pliers, pull up any carpet tacks left in the wood. Be careful not to splinter the wood. Thoroughly vacuum the stairway.
Measure one of the steps, starting at the top of the tread and moving across the tread to measure the height of the riser as well. Add 1 inch to your measurement and multiply that measurement by the number of steps, not counting the last riser before the landing. Measure the last riser and any landings that will be carpeted with the stairs, and add one inch for turning under. Add this to the first number and divide the total by 36 to determine the number of yards of carpet runner needed. You will need approximately the same amount of padding.
Buy the carpet runner width that best fits your stairway (it is usually sold in 27 and 36 inch widths). Don't use leftover carpet, because then you'll have to turn it under on the sides and that can get tricky. Buy high quality heavy felt or rubber stair carpet padding. Ask the carpet dealer to help you figure out the amount of padding and number of tackless fastening strips you'll need. Also, make sure to rent a knee kicker from the carpet dealer; this is used to stretch the carpeting tightly over the stairs and the fastening strips.
Put on your work gloves. Nail fastening strips at each riser tread corner. Measure the width of your stairway, and subtract the width of the carpet runner from that measurement, then divide by two. This is the number of inches that you need to measure in from each side of the stairs to make sure the carpet runner is centered. Measure one side, and mark with chalk. Measure on the other side, and mark that as well. Do this at the base of each riser and the inside of each tread, and then measure between the chalk marks to make sure it matches the width of the carpet runner.
Cut the tackless carpet fastening strips to the width of the runner using a small handsaw. On every stair, nail a strip centered on the riser, with the teeth pointing down, 3/4 inches above the surface of the tread. Use a 3/4 inches piece of scrap wood to hold the strip in place while nailing. Nail another strip centered on the tread, 5/8 inches out from the riser. Do this for every step, but don't nail a strip onto the top riser.
Measure and mark the carpet padding. You want it to be as wide as the carpet runner, less about 1/4 inches, so that it fits neatly under the runner. Using heavy duty scissors, cut a strip of padding large enough to fit over each stair tread, long enough that it goes from the edge of the fastening strip to two or three inches onto the riser below the tread. Staple the padding in place, waffle side up, setting staples in at a diagonal angle at every 2 inches along the edge. Start with the side by the fastening strip, and staple the other edge while holding the padding tight. Do this for every step.
Unroll the carpeting over the stairway and make sure that the nap is lying downward across the stairway. Position it from the bottom up, making sure that the carpet is positioned straight over the fastening strips and between the centering marks. Trim any uneven edges along the bottom with a utility knife. Push the point of an awl into the carpet at one side and use it to push the edge of the carpet into the fastening strip. Work across with the awl and smooth the entire edge into place. Finish the bottom by wedging the loose carpet end into the 3/4" gap between the fastening strip and the floor. Use a tack hammer to drive the carpet firmly into place.
Stretch the carpeting up and over the first stair using the knee kicker. Pull the carpet runner into place, making sure it's straight. Set the head of the kicker in the center of the tread, about an inch from the tread riser edge and pointed directly at the riser. Kneel on the floor and hold the kicker in one hand and a stair wedging tool or broad bladed chisel in the other hand. Put all of your weight on one knee and then bring the other knee forward to hit the cushioned end of the kicker. This stretches the carpeting under the head of the kicker to hook it onto the teeth of the fastening strip. While working the kicker, push the carpeting into place using the wedging tool or chisel as a lever. Work from the center of the tread out to the sides. When the carpet has been kicked and folded into the corner along the step, wedge it permanently into place along the riser-tread joint, using a hammer to drive the wedging tool or chisel into the corner. Repeat the procedure to cover each stair.