Select unwashed potatoes that are firm and dry. Avoid any with green patches - the green is an indication of the toxin solanine. Avoid also potatoes that are wrinkled and spongy.
Old potatoes will keep for weeks in a paper bag with plenty of ventilation.
Never store potatoes in plastic bags.
New potatoes should be used within a few days.
Tips & Tricks:
For lower GI buy waxy potatoes.
Floury potatoes (Bison, Sebago) are suitable for mashing and baking (and chips, although no-one reading this site will be making chips!)
Waxy potatoes (Desiree, Kipler, Pontiac) hold their shape and are suitable for boiling and salads.