How important is it to wash your hands?

Extremely. Hand washing reduces bacteria that can enter the body thru the eyes, nose, ears, mouth, skin, and other openings. Therefore, it will reduce the chance of infection.
Very. The #1 cause of infection spread anywhere in the world in any setting is germ transfer from the hands. You can spread infection to other parts of your body by touching it (e.g. Your face), from person to person, and from object to yourself. Handwashing significantly reduces this.
Very important. Washing of hands reduces the transmision of disease.