Installing baseboard around stairs certainly gives you the opportunity to use the miter saw. It takes much longer to install baseboard around stairs rather than walls, simply due to the number of cuts and the short pieces you will be using. With a wall, you may be able to do 20 feet in a matter of minutes. With stairs it may take you a few hours.
Installing baseboard on walls, the miter cuts are in a vertical position. With stairs, the miter cuts are as the baseboard is laid on the wall so that the entire piece changes direction to follow the stairs. Most pieces will have two miter cuts with the exception of the very top piece and the very bottom piece. These pieces lay in a horizontal position and either begin or end the directional change.
The cuts will be at 45-degrees and will mate with the corresponding 45-degree cut to make a 90-degree corner. You will need to map out which direction the cuts are to be made and for this I would recommend making some templates out of scrap material. That way you can place two cuts together and make sure they are correct before you use your nice stock. You can start out using a measuring tape, but you will likely have to nibble away with a saw during the installation.
I would start at the top, cut the first piece, and nail it in place with a nail gun. Then move to the second piece and nail it, and so on. I would not cut all the pieces at one time and try to install them as there will be some variation.