Beware the refrigerator
Stone fruits—like peaches and nectarines—are seasonal fruits, with a short growing season and even shorter shelf-life. That's why it's so important to learn how to store nectarines and other stone fruits correctly. Figuring out how to store nectarines and peaches properly can make the difference between biting into something that's mealy and grainy, and actually enjoying a delicious piece of fruit. The trick to properly storing fresh stone fruit is to make sure they don't get too cold before they're fully ripened. That's because preemptively exposing these fruit to cold is what causes graininess.
As Harold McGee writes in On Food and Cooking, peaches and nectarines "tend to become mealy or break down in prolonged cold storage," and this is part of the reason they're more seasonal. You just can't keep peaches and pears in cold storage for a long period of time without sacrificing quality. Apples, however, can stay in cold storage for up to a year without any real issues. So if you get a mealy peach or nectarine from the supermarket, the reason is probably because it was stored at too low of a temperature, below about 45°F, according to McGee.
That's why, as a general rule, you shouldn't store nectarines or peaches in the fridge until they are fully ripe. And you'll know when a peach or a nectarine is fully ripe by the fragrant smell and the feel of the flesh. It'll be soft but firm and give a little bit when you press it gently.
If you have picked up an unripe nectarine or peach, store it on your counter at room temperature, ideally away from sunlight. That's a process that usually takes a day or two. If you want to eat an unripe peach sooner than that, you can pop it in a paper bag for a bit; that'll speed up the process by trapping ethylene, the plant hormone that causes ripening.
So only store stone fruits in the fridge once they're fully ripe, otherwise you'll end up with mealy fruit. And even once you've got fresh, ripe nectarines and peaches in the fridge, you're really only able to extend their shelf-life by a day or two. As with most fresh fruits, the best way to store peaches and nectarines is to eat and use them as soon as they're ripe.