Weekly: Take your broom outside and gently bang it against a tree or the side of the house to remove loose particles. Or clean the bristles with your vacuum cleaner.
Monthly: Place the broom (whether synthetic or straw) in a bucket of warm water with a few drops of dish soap added; let soak for an hour. Rinse and let dry for several hours before putting it away. For extra cleanliness credit, give the dustpan a once-over with a bleach wipe.
Tip: Consider designating a special broom for each germy area (the bathroom, around the litter box, etc.) and another broom for the kitchen.