Store vegetables away from fruit if possible. Some fruits produce ethylene gas which causes other fruit and vegetables to ripen more quickly. Even in the fridge, if you have two crisper drawers, use one for storing fruit and the other for vegetables.
Most vegetables prefer a little moisture, while most fruit prefers less humidity. By closing the vents in the vegetable drawer, you will help maintain more moisture and vegetables will stay fresher for longer.
In addition, remove vegetables from any plastic packaging to help prolong freshness and nutrients. Vegetables like air and allowing it to circulate freely will help them breathe and prolong freshness.
Some vegetables prefer to be kept in the dark. Potatoes and onions should both be kept in a cool, dark cupboard, away from each other, as onions produce moisture which make potatoes sprout more quickly, and away from any heat source, which also speeds up sprouting.
I find the best way to use up vegetables is by making various soups, even during the summer months my children love a bowl when we get home from the beach or a day out. They can also be added to smoothies, used in a stir fry, or you could blanch and freeze them for use at a later date.