First of all, don’t squeeze the peaches! Squeezing damages them and results in bruising. You can tell if peaches are ripe by sniffing them; when they’re ripe, they smell… like peaches. Don’t be afraid of buying the hard-looking, unripe peaches. Be gentle and do not bang or dent them in the car or on the way home.
Place the peaches in a brown paper bag or between the folds of a linen towel on the counter. The peaches give off ethylene gas; when trapped, the gas helps the fruit ripen. (A plastic bag will work, but because plastic doesn’t breathe, there is more danger of the peaches rotting.) Check the peaches every day by opening the bag/towel and sniffing; sometimes they ripen within a day.
Once ripe, peaches can be refrigerated if you cannot eat them fast enough. Wash peaches just before eating.