Wound care. Washing a cut or scrape with soap, and water and keeping it clean and dry is all that is required to care for most wounds. Cleaning the wound with hydrogen peroxide and iodine is acceptable initially, but can delay healing and should be avoided long-term. Apply antibiotic ointment and keep the wound covered.
Simple rules. The real question is what not to do that slows down healing. Avoid using alcohol, peroxide, hibiclens, betadine, and bleach solutions. Inflammation (red, hot, swollen, tender) is important to healing so anti-inflammatories (motrin, advil) are bad. Cover the area with a clean dressing changed every other day and protected from trauma. Topical antibiotics help, oral antibiotics do not.
Local hygiene. Wash the area with soap and water. Apply antibiotic ointment and cover the wound. Do not let it dry out.