Most nicks, scrapes, cuts and sores heal themselves naturally in a couple of days, but if you have a larger abrasion, or a weakened immune system, you need to take extra care.
The best way to heal cuts is to keep it from becoming an infected cut. Once you receive an injury, assess the amount of damage. Depending on the severity you may want to consult a doctor about stitches or an x-ray to check for fractured or broken bones. If you intend to treat it yourself, make sure you clean it out thoroughly. Soap and water can be effective, and even if it hurts use a clean cloth to scrub out any dirt, sand or other foreign material. If you are not capable of doing this yourself either due to reach or not being able to cause pain, find a friend or see a doctor.
Next, treat the wound with something that kills germs. Rubbing alcohol, iodine, and hydrogen peroxide do the job okay, but may irritate surrounding skin, depending on your sensitivities. Antibiotic creams can be effective, but they contribute to bacteria’s growing tolerance to antibiotics. Whichever you choose, apply it liberally to the site and surrounding tissue, either by dribbling it directly or with a bit of gauze or a cotton ball.
Once the cut is cleaned, apply a bandage (make sure to dry non-injured skin the bandage will need to stick to). If there’s severe skin damage, consider putting a barrier between the cut and the bandage (sterile products such as wire mesh are available alongside bandages at your local store) that prevents the bandage from sticking to the cut. Air helps the wound to dry and scab over, so choose a bandage that allows some airflow but still prevents other dirt and pathogens from getting in unless you have a specific need to prevent water from entering, and then switch bandages only for that occasion. To help prevent infection change the bandage frequently (use your judgement based on the healing of the wound: how quickly is the bandage getting dirty?), reapplying whatever topical ointment you used.
If you see signs of pus, swelling, or redness you may have an infection. Consult a doctor.