Where to store fruit and vegetables varies between person to person and generation to generation. To get the most out of your produce, store most vegetables in the high-humidity drawer in your refrigerator. Most of your fruits should be stored in the lower-humidity drawer.
Don't mix your fruit and vegetables together. The gases from one type of produce may cause the others to ripe much more quickly. I have a separate basket for my bananas and onions that I store on my counter.