The first and possibly most important step when treating minor cuts and scrapes is to thoroughly clean the wound with mild anti-bacterial soap and water. You can use sterilized tweezers to remove any debris that remains embedded in the wound after rinsing. This will reduce the risk of an infection and possible complications. If the debris is abundant or can't be removed for some other reason, a trip to the emergency room will be necessary.
Water may induce bleeding by thinning the blood. If while rinsing the wound you notice increased blood flow, use gauze or a clean cloth to apply gentle, continuous pressure until the blood clots.
Although hydrogen peroxide is commonly used as a disinfectant for minor cuts and scrapes, it is actually not very effective and may even delay the healing process by irritating a person's living cells. You can use hydrogen peroxide but apply it around the open wound, not directly to it. An antibiotic ointment such as Neosporin is a better alternative - it will keep the wound from getting infected and speed up the healing process.
Dress the wound with a bandage or sterile gauze to keep dirt and bacteria out. Change the dressing frequently and rinse the wound as often as necessary to keep it free of dirt.
Continue to monitor the wound for several days. If the injured area turns red or puffy, or if excessive pain persists, then it may've become infected and will require a physician's attention.
Do not simply dismiss cuts as minor injuries as some may be quite serious if not tended to by a professional. If the wound is very deep or the bleeding is profuse, it may require stitches in order to heal properly.