Potatoes are live tubers that contain all the genetic material required to grow a new potato plant. Keeping potatoes fresh is different from keeping carrots, beets or root vegetables fresh. Those veggies are progressing toward death. Keeping potatoes fresh is another story. You're actually keeping them alive.
Brush off the dirt from the potatoes with your hands or a paper towel. It's not necessary to wash the potatoes.
Take a hole puncher and punch holes into a plastic bag. The plastic keeps the humidity level high so the potatoes won't dry out.
Place the loosely filled bag in a dark cool place. Light encourages the potato to sprout which ruins it. The best storage location will be in an unheated store room, garage or mud room. The ideal temperature is between 45 to 55 degrees. Keep the potatoes above freezing.