Almost all fruits and vegetables produce invisible and odorless ethylene gas, "the ripening hormone." Fruits that produce ethylene gas in larger quantities ripen faster as a result of concentrated exposure. Other fruits aren't as affected by ethylene gas, and can be stored with apples and other ethylene-producing fruits. The issue is more complex than simply sorting your fruits, though. Exposure to ripe or overripe fruits can cause vegetables to rapidly decay.
Most fruits can be safely stored together, though one piece of overripe fruit will cause the rest to go bad more quickly. By refrigerating some high-ethylene fruits, you can slow their ripening and enjoy them longer. Fruits that can be refrigerated include apples, apricots, cantaloupe, figs and honeydew melons. Contain them in perforated plastic bags or breathable storage containers to keep them from exchanging aromas and flavors; apples are major offenders here. You can safely store these with fruits such as cherries, blueberries and citrus, but keep them well away from watermelons.
Some high-ethylene gas fruits are sensitive to cold, and if refrigerated before they're fully ripe their moisture levels and flavor are greatly affected. Once they're fully ripe you can refrigerate most of them -- but warm them to room temperature again before eating, for full flavor. Fruits that fall into this category include avocados, unripe bananas, nectarines and peaches, pears, plums and tomatoes. Don't store ripe bananas in the frig, though. Their skins will turn dark -- making them look spoiled, though they aren't -- and everything in the vicinity will smell like bananas.
Fruits and Veggies
Keep lettuce, fresh greens and almost all other vegetables away from fruit, which generally produces more ethylene gas. Making this a habit will make it easier to keep your produce fresher longer. In addition to lettuce and all leafy greens, you should segregate salad vegetables including carrots, cucumbers and peppers, and all the cole vegetables: broccoli, Brussels sprouts, cabbage and cauliflower. Also keep eggplant, parsley, peas, potatoes, squash and sweet potatoes separate from fruit.
Take advantage of what you know about ethylene gas to naturally speed up ripening of various fruits. Encourage faster ripening of peaches or pears by placing unripe fruits together in a closed paper bag on the kitchen counter. The bag still "breathes" enough to avoid suffocating fruit, which plastic would do, but it concentrates ethylene gas. You can "speed ripen" a single unripe peach by placing it in a paper bag with a ripe banana or apple. Keep in mind, though, that quicker ripening is actually a form of accelerated aging. If you don't monitor ethylene-enhanced ripening carefully, your fruit may rot before you get the chance to enjoy it.