Potatoes keep best when placed in a well-ventilated container and stored in a dry location, away from sunlight, and at temperatures between 45 and 55 degrees F. That isn’t too easy to find in most homes today but if you place them in a paper bag, cardboard box, or bowl (not in a plastic bag) and keep them in the coolest part of the kitchen or a dry part of your basement, it should help their longevity. If you are shopping for a small family, buy only enough potatoes for a week at a time. Refrigerator temperature is a bit too cool for potatoes and tends to increase their sweetness making them brown very quickly when fried.
You can find a lot more information on all aspects of handling potatoes at potatogoodness.com. Scroll down to the bottom of the page and select a category. In addition to nutrition information and recipes meant for home cooks, the categories intended for professional cooks and markets are pretty interesting too.