What is the best way to care for my son's skin after circumcision?

Soap and water. Just keep it clean with baby soap and water between diaper changes. Don't run with baby wipes as that can be painful.
Vaseline, a few days. After the "foreskin" has been cut off, the wound area is a little oozy with a slight amount of blood and sticky material. For about 3 days, the parent can put some vaseline on the wound so the end of the penis doesn't stick to the diaper. Also, put some 4x4-inch gauze cloth around penis before closing the diaper (so less or no poop will get on the penis). After 3-4 days, no special care needed.
Petrolatum. Wash and wipe around penis till healed. Remove poop if necessary. Use lots of vaseline until it has healed. Later on look out for meatitis which is redness and inflammation around the urinary opening, this can lead to meatal stenosis (narrowing). Leave him out of diapers in crib if there is redness, allow air to get to it, and use Bacitracin or vaseline when he has to be put into his diaper.
Gently. The area needs cleaned with lukewarm water until healed. Covering the area with a gel such as petrolatum jelly, vaseline, or a &d ointment soothes and protects the glans from rubbing.It often can take several weeks for it to totally heal. When the color becomes flesh tone, that usually means it has healed.
A&D ointment. Until the penis has healed which usually is only two or three days vitamin a&d ointment should be applied either directly to the penis or generously applied to the front of the diaper to prevent the healing wound from sticking to the diaper.