Ice for 1st 2 days. In general, ice will help decrease swelling and pain for the first 2 days. Topical antibiotic ointment will prevent infection if blistering occurs. If just red like a sunburn, don't worry. If you have no sensation of the burn area, it may be a deep burn (deep enough to damage nerve endings). If so, get to see your doctor to make sure you don't need more aggressive care than above.
Quickly. Run cool not cold water over the burn for at least 10 minutes. Do not put ice on it; actually causes further damage. Burned skin is very fragile, and you can even end up with frostbite on it. Then pat the burn dry and, if it's a minor burn, bandage it lightly with nonstick gauze and if it's very bad (large blisters, charring, covering a large area go to the E.R.
Depends. Treatment depends on depth. Most burns on the palm of hand (from grabbing a hot pot) are either first or second degree burn (blistered). Both of these are best treated initially with cold water, then apply an ointment on the blisters to keep the air off and the burn moist. Avoid rupturing the blister. If ruptured even more important to keep antibiotic ointment on. See doc if signs of infection.