Minor cuts and scrapes are generally more of an annoyance than a serious health problem. They can cause serious discomfort, especially if they appear on a part of the skin that is easily irritated. Once a cut has stopped bleeding, the best way to speed the healing process is to keep it from getting infected. This is because while the body is fighting an infection, it has to slow the healing process.
Clean the wound. Cleaning any dirt and debris out from a cut will help it heal faster. Rinse the wound out with clean water. Don't use soap, as this can irritate the wound. If after rinsing, any debris remains, dip some tweezers in rubbing alcohol and use them to remove anything that could irritate the cut.
Apply an antibiotic cream or ointment. Although these treatments don't actively speed up the healing process, they do prevent infection. This is important because an infected cut will be more painful and will take longer to heal.
Cover the cut with a bandage. This will help keep bacteria out.
Remove the bandage once the cut begins to heal. Once the cut has healed enough that it is unlikely to become infected (when the cut has mostly closed up), leaving it exposed to the air will help it heal faster.