Immediately after the burn, run it under cool-lukewarm water. Not cold, not hot. You may or may not get a blister. If you do, don’t pop it. Simply bandage it with nonstick wraps, the liquid in the blister helps your skin heal. If it pops treat it like an open wound. Aloe is good to use. It can sting like shit if the burn is too fresh, but does help. Better yet is an aloe based burn gel. After initial healing, keep it moisturised as you said.

Not a doctor or medical person, but I am a very experienced fire performer and have tended to all sorts of burns.