What's the fastest way to heal an abrasion?

Keep it clean: Thoroughly clean an abrasion, remove any foreign material, apply a topical antibiotic cream if wound was dirty or is deep.. Sometimes it is helpful to cover the wound to keep it clean but many small abrasions can heal well when exposed to air as long as they are kept clean.

Air is bad : Gotta disagree with prior answer. If you remember that wounds heal by growing healthy moist tissue in from the bottom or by the elastic sides pulling things in from the sides, then having a scabs form slows down healing. Open wounds should be exposed to air for only the shortest time possible. Cover with clean dressing changed avery opther day and protect it from trauma will provide healing.