Do Not Store Fruits and Vegetables Together. Fruits that give off high levels of ethylene (the ripening agent) can prematurely ripen and spoil surrounding vegetables. (Think of the "one bad apple" adage.)
For Vegetables: Before storing, remove ties and rubber bands and trim any leafy ends. Leave an inch to keep the vegetable from drying out. Make sure the bag you store the veggies in has some holes punctured to allow for good air flow. Pack vegetables loosely in the refrigerator. The closer they are, the quicker they will rot. Leafy greens can be washed before storing by soaking them in a sink full of water, while soft herbs and mushrooms should not be washed until right before they are used.
For Fruits: Non-cherry stone fruits, avocados, tomatoes, mangoes, melons, apples, and pears will continue to ripen if left sitting out on a countertop, while items like bell peppers, grapes, all citrus, and berries will only deteriorate and should be refrigerated. Bananas in particular ripen very quickly, and will also speed the ripening of any nearby fruits.