Mushrooms need to be kept cold in a breathable bag, so store them in the fridge in the paper bag they come in. Shiitake mushrooms have a tough stem, so remove that and save it in the freezer for stocks. Oyster mushrooms are all edible, except for the dirty little bit at the "root." Oyster mushrooms are more delicate than shiitakes, and need to be eaten within a week or less. Shiitakes tend to dry out, rather than mold or rot (depending on the moisture content), and if they dry out, you can still use them. Just revive in hot water or add to soups. Rinse fresh mushrooms quickly to remove debris, and allow to drain before cooking.