Installing carpet on stairs and landings is not any more difficult than installing carpet anywhere else in your home. Carpet the landing part of the stairs just like any other flat surface. The stairs can be covered using a runner that is either as wide as the steps, or a little less wide, depending on the look you're going for.
Install a rug pad on the landing by driving a staple along the edges of the rug pad; each staple should be about 1 inch apart from the next staple. Cover the entire landing with rug padding.
Install rug padding to the outer corners where the threads and risers meet in the same manner as you installed rug padding to the landing. The thread of a step is the part of the step that you step on when going up or down the stairs. The raiser is the part of the step that goes up to the next step, down to the floor, or up to the landing--basically the part of the step perpendicular to the thread. The section of rug pad on the thread should reach up to only 1 inch from the next riser, and the section of rug pad on the riser should only go down 1 inch on the riser.
Install a tackless strip in the sections of the thread between the rug padding and the next riser, then install a tackless strip on the riser where the riser and the next thread meet so that you have two tackless strips on each inner corner of the steps. Tackless strips have nails sticking out of one side; to install them, hammer in the nails to the wood of the threads and risers.
Install carpeting on the landing by hammering carpet tacks to hold the carpet in place; each carpet tack should be about 2 inches apart from each other. Cover the entire landing with carpeting. Tuck in all the corners using a blunt chisel; place the chisel onto the carpet at the edges of the landing, then push that carpet into the corner created by the floor and wall.
Cut out a runner of carpet that is as long as the slope of the steps. Use a carpet cutter to cut the runner.
Attach the runner to the place where the last riser of the steps and the landing meet. Place carpet tacks running along the edge that the riser shares with the landing, positioning the tacks about 2 inches apart from each other.
Pull the runner down and hold it flat against the riser until you get it to the tackless strip, then push it down flat onto the thread of the first step. The tackless strips will hold the carpet runner in place. Repeat this step on each step until you get to the bottom.
Attach the runner to the last riser of the steps using carpet tacks the same way you attached the runner to the top of the first riser.
Cut away any excess rug with the carpet cutter, then tuck in the bottom of the carpet into the bottom of the steps with a blunt chisel.