Treating a minor cut or scratch is mostly about keeping it clean, but the only thing you really need to clean it with is water. Regular tap water works just fine. If there's a possibility it could get dirty, cover it with an adhesive bandage (Band-Aid or some other brand). Most importantly, as a cut is healing, rinse it with tap water whenever it gets dirty.
Using triple antibiotic ointment could help some abrasions (scratches) heal a little quicker and probably with less pain at first, but keeping the dressing fresh and moist is more work than the average scratch is worth. Putting triple antibiotic ointment on a wound could also lead to redness, itching and burning -- what doctors call allergic contact dermatitis -- which looks a lot like an infection.
Walk through this with me. You get a cut and clean it appropriately, then cover it with triple antibiotic ointment. After a day or so, the wound and the surrounding skin get painful and red. Thinking your cut is infected you put more triple antibiotic ointment on it, which makes the dermatitis worse.
A minor wound just isn't worth the headache.
Trust your body to heal the way it's supposed to. If you have some sort of medical condition that keeps your body from healing correctly, then talk to your doctor about the best way for you to handle minor cuts and scratches.