Hi Linda, welcome to DC !
To speed up the ripening process in peaches, place them in a loosely sealed brown paper bag on the counter. The peaches give off ethylene gas - a natural ripening hormone - which is trapped inside the bag causing the fruit to ripen faster. You can do this with a plastic bag too, but the plastic doesn't "breathe" as well as the paper, meaning moisture may collect causing your perfectly ripened peaches to rot.
Once ripe, peaches should be refrigerated and used within a few days (although they'll taste better at room temp). Ripe peaches have a sweet fragrance, are soft (but not mushy) to the touch and have a golden undertone.
This ripening trick also works for apricots, nectarines, avocados, bananas, kiwi, mangoes, pears and plums.