Most fresh fruits and veggies don't have a very long shelf life so only buy what you're going to eat within a few days. Most will be just fine in the crisper drawer in the fridge, with a few exceptions.
Take your hints from the grocery store displays. If the store has it in a refrigerated case then refrigerate it at home. Hang bananas from a hook so they don't bruise from laying on the counter. I've even used an old bread twisty-tie and hung them from a cabinet knob.
most veggies and fruit lose a lot of flavour when chilled, something about the sugars binding up or something, my pops in law tried to tell me about it and tomatos, he grows them and is super picky about how they get stored. also apples and maybe some other fruit tend to release ethylene as they ripen, causing other things to ripen and over ripen, choose bedfellows carefully. I think along RJS's lines, get a couple three days worth if possible, eat and replace.