Peaches will soften in 1-2 days on the counter. To hasten the process, put them in a brown paper bag with a banana or an apple; watch carefully because they can go from ripe to over ripe very quickly. Use within a day or refrigerate. Use within 4-5 days from the refrigerator. Peaches will continue to slowly soften in the refrigerator and they will gradually dehydrate, so you do not want to leave them in the fridge too long. You are better off putting your peaches in the fridge, and then putting them on the counter to ripen rather than putting ripe peaches in the fridge. Do not wash your peaches and then store them - the water encourages rotting.
As we come to the end of the peach season, it is especially important to follow the above guidelines. End-of-season peaches will not get as soft as mid-season peaches. If you are waiting for soft peaches, stop; they won't happen - you'll get mealy peaches instead. Leave the peaches out for 1-2 days and then eat.
MELONS
Whole watermelon will keep in the refrigerator for 4-5 days. Cut watermelon pieces will keep in the refrigerator for 5-6 days. Whole watermelon will keep unrefrigerated for 1 day.
Because of the excessive rain this year, melons will not keep the way they have in prior years. Also remember that melons soak up a lot of water, so since this season has been rainy, they will have more water, aka, the flavor will not be as intense and the flesh will not be as firm as drier seasons.
APPLES
Apples keep best at 33 degrees believe it or not. Since most people don't have cold storage options that stay at 33 degrees, the fridge is your best bet. Apples will keep in the fridge for about 4 weeks, depending on the time of year and the variety (the harder the apple and the less storage time before reaching you, the longer the apples will keep).
The weather has not been kind to the apples this year. Their keeping quality is going to already be compromised by the time they reach you, simply due to the weather. So be nice and keep them refrigerated :)
Basically, when in doubt, refrigerate!