Storing your Hat
It's best to place your hat in a hat box in a cool, dry area, but if you don't have one handy, then your next bet is to either lay your hat upside down on it's crown in a safe place (I keep mine on the top of a cabinet), or hang it on a hat rack or peg. Make sure the hook is wide enough to not create dimples in the hat.
Both laying your hat on it's crown or hanging it on a rack for long periods of time can tweak the shape of the crown or brim. So like I said, a hat box is the best bet.
For loose dirt, take a horsehair brush or hat sponge to it. Make sure to use a brush with the same color of bristles as the hat, (black for black hat, light for white hats).