Asparagus & celery: Wash & cut off ends. Store in a container with stalks standing upright with a little water on the bottom.
Bananas: Buy small bunches in varying stages of ripeness. Break up bunches and store on the counter in a container that allows for good air circulation. I have a wrought iron basket that works really well.
Berries: Store in fridge. Berries just don't last very long, period, unless you opt to freeze them. However if you wash them, they spoil faster, so opt to wash just those that you'll eat immediately. Also be sure to pick out any moldy berries immediately as they will cause others to spoil faster.
Broccoli & cauliflower: Will stay fresh in the fridge for about 10 days. If you opt to freeze, you'll need to blanch them first which is a very easy process detailed here.
Green onions: Slice them and store them in a container with a folded up paper towel. They'll stay fresh for about 10-12 days this way.
Lettuce & spinach: Wash & dry leaves. Store leaves in a loose but closed bag or container with a folded up paper towels. The towel absorbs the moisture and leaves stay fresh longer.
Mushrooms: Store in the fridge in a paper bag. Will stay fresh for about a week this way. {Keeping them in the container you bought them in makes them slimy within 2-3 days.}
Onions: Will stay fresh in a cool, dark place for 3-4 weeks. You can also dice them and freeze them for use beyond that time frame.
Pears: Store in fridge until ready to eat. Once they're out on the counter, they'll ripen within 2-3 days.
Potatoes: Open bag they came in and store in a cool, dark, dry place. I store mine in a cupboard in my garage- they keep for 5-6 weeks!
