This site has a rundown of where to store your fresh produce, and includes a handy chart you can print and stick to the fridge.
I have learned that I've been doing some fruits and veggies wrong. For instance, all fresh herbs except basil go in the fridge--ah ha, no wonder the basil kept turning black in there. Also, I tried her suggested way of keeping herbs fresh: snipping their ends and putting them in a glass of water, like flowers in a vase, then slipping a plastic bag over the top and refrigerating them. The cilantro I bought has stayed fresh for over a week now with this method--much longer than it does when trapped in its original plastic bag in the produce drawer.
Others I've been wrongly refrigerating when they should be on the counter: peppers, cucumbers, and tomatoes. (Until you slice them up at least, presumably.) Meanwhile, plums and cherries should go in the fridge, when here I'd been storing them in the fruit bowl. They did seem to rot awfully fast that way, come to think of it.
Furthermore, keep those darn apples and pears and bananas away from other produce: they emit ethylene, which ripens other fruits and vegetables at an accelerated rate. However, you can use that to your advantage if you want to, say, ripen an avocado quickly for your guacamole.
Anyway, only a chef or a Food Sciences major could keep all this straight, so print the chart and enjoy your happy produce.
(And happy Memorial Day!)