Peach season depends on the area you are talking about, but for California peach season starts around June.
In places like Michigan peach season begins around July or August and continues on from there. When it comes to California peaches, they are generally available from the month of April through October, peaking in July and August.
When you are shopping for peaches, you will want to look for ones that are very plump and anywhere from medium to large in size, so you will get the most out of the money you pay for them.
As far as storage goes, you will want to handle them carefully and put them in a cool place until you are ready to eat them.