The following fruits are best stored in the fridge:

Cold temperatures will help keep apples and pears crisp, and help prevent citrus from molding, grapes from becoming soft, and soft fruit from rotting. If you don't like eating cold fruit, try taking a few pieces at a time out of the fridge to store at room temperature.

If fruits like apples, kiwi, melons, nectarines or pears are especially unripe, you may want to store them at room temperature for a day or two to speed up the ripening process. If you want fruits to ripen faster, put them in a plastic bag with a banana, and put the bag on top of the fridge for a couple of days. Bananas produce ethylene gas, which speeds up the ripening process of other fruits. The heat from the fridge stimulates the ethylene production, and the plastic bag traps the gas to speed up the ripening process for other fruits.

These vegetables are very sensitive to moisture, and will either brown ("rust") easily or become soggy and limp. Your best bet is to simply eat them quickly, but if you must store them, try putting them in brown paper, then plastic, in the fridge.