It's important to store peaches properly to keep them tasting fresh. In the case of ripe peaches, they do well on a kitchen counter if they're going to be eaten within 24 hours.
Leave them out of direct sunlight, stem-side down to avoid bruising, on a non-porous surface such as glass, stainless steel, or porcelain to keep them fresh for a day or two.
Store unripe peaches in a paper bag on the counter, which speeds up the process. If they're a little too ripe, keep them in the refrigerator, stem-side down, for two or three days.
Keep them unwashed, whole, and outside of a container or bag. The cool environment will prolong the peaches' lifespan, whereas overripe ones on the counter may last half a day.
If you've picked more peaches than you could possibly eat in a day or two and need to store them longer, your refrigerator or freezer is your best bet.
For freezer storage, peel your peaches, then slice them into wedges. Freeze in a single layer on a plate or baking sheet first, then transfer into a large resealable freezer bag.