Try not to put them in at all unless you have to. Of course once ripe, they store best in the fridge. Use split tomatoes at once since they are more prone to rot and can harbor harmful bacteria. Papayas, mangoes and avocados also taste better if they are fully ripe before you refrigerate them. If you want to put them in the fridge, place them in individual paper bags first.
Bananas are one fruit that ripens well off the plant.
They are also very sensitive to cooler temperatures. Ripen at room temperatures and then eat them immediately. Freeze of refrigerate the rest for cooking or baking. Store Irish potatoes and sweet potatoes in a cool dry place, but not the fridge. Irish potatoes stored in the refrigerator often develop a sweet flavor as the starches turn to sugars.