Store your cowboy hat with the brim up. This protects the brim of the hat from warping, which occurs when the hat rests on its brim on any flat surface for an extended period. You could also hang your hat on a hat tree or from the knob of a coat tree.
To pick up your cowboy hat grasp its brim. If you pick it up at the crown, you will end up further indenting its crease, much like the Old West cowboys first caused dents in their round crowned hats.