Vegetables need good air flow! Don't keep them in wrappers or plastic bags as this will make them deteriorate quickly. Likewise when storing them in your fridge you shouldn't pack them in tightly as they will rot quicker. Here is a simple guide of what to store in the fridge and what to store in the cupboard - by this I mean a cool dry place. Not freezing but not in direct sunlight!
Fridge - Mushrooms, herbs, leafy greens (kale, spinach, watercress) broccoli, asparagus, carrots, parsnips
Cupboard - potatoes, garlic, onions
Fruit and veg shouldn't actually be stored together. Fruits give off high levels of ripening agents and this will spoil any surrounding veg. Bananas especially should not be stored by other fruit or veg as they ripen super quick and will do the same with surrounding fruits again because of the high level of ethylene they give off.
We always go by the rule it is has a stone in the middle it needs to be left out on the countertop (fruit bowl) to ripen. Whereas berries, grapes, citrus will only deteriorate and they need to be in the fridge. As I mentioned bananas ripen very quickly so if you want to speed up the ripening of other fruits keep bananas close. If not they should be kept separate. Soft fruits will not last even a day if kept out of the fridge. As soon as possible they need to be refridgerated or consumed. There not a fruit with a long life whether you keep them in the right conditions or not.
Here is a guide where to store fruits.
Fridge - Berries, oranges, lemons, grapes, grapefruits.
Fruit bowl - bananas, apples, pears, avocado, pineapple, mango