Please advise what is the quickest way to heal a cut (minor)?

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. Antibiotics do not help healing unless it is truly infected.