Most cuts are not very deep. Cuts that need stitches are deep or leave the skin edges separated. Also, deep cuts that are longer than 1/2 inch (1/4 inch if on the face) need to be closed by a doctor.
Apply direct pressure for 10 minutes to stop any bleeding. Wash the wound with soap and water for 5 minutes. Apply an antibiotic ointment. Cover it with a Band-Aid or gauze. Wash the wound and change the Band-Aid every day. Instead of a Band-Aid, you can use a liquid skin bandage to seal the cut. These are available at your local pharmacy. Give acetaminophen (Tylenol) or ibuprofen (Advil) as needed to help the pain.
Don't use alcohol on open wounds. It will sting and damage normal tissue.
Note: Deep cuts must be closed by a doctor within 12 hours of the time of injury. They are much less likely to get infected if they are closed within 2 hours.