With any first aid situation, assess the scene make sure it is safe. Then start the primary survey as follows...

ABC's:

Airway

Breathing

Circulation

Firstly, run the burn under cool water for 15 minutes. Do not put ice on it, for it may cause tissue damage.

In the first aid kit, (or some of them anyways) you will find dressing specifically for burns. Never use gauze, or toilet paper. This will stick to the burn. You may want to wet the bandages or dressings before applying. Apply securely on the burn, call for medical help.
Put a very cold ice bag or cold cloth over the burn(s) and have the person rest for a few hours. Put some Aquaphor (petroleum jelly) AROUND the wound to soothe it.

Alternate Advice (A little gruesome for the squeamish, but it's a gruesome topic...if easily upset stop reading NOW.)

Use COOL sterile water, not ice bags, to gently flush the burn, to literally bring the temperature of the burned area down so the physical burning stops. Just like a steak will continue to cook from residual heat when taken off a BBQ grill, so will your flesh if the burn is from heat such as a flame. Then LIGHTLY put sterile dressing gently over the burn, to help shield the burn area from air flow, which can causes additional pain. In extreme cases you might pour some sterilized water over the sterile dressing for partial temporary pain release (to some minor degree. ) Seek medical attention as soon as possible.

Do NOT use "a very cold" ice bag: You can actually cause MORE damage to the area by 'freezing' it, and accidently causing frostbite on top of your burn.

Keep all additives such as petroleum jelly or butter away from the burnt area. In the case of a severe burn, hospital personnel will have to remove any foreign matter from the burn area to help prevent infection. If you have put petroleum jelly (or butter!) onto the burnt area, this may cause much additional pain as the medical professionals later need to scrub the area to remove the ointments! This is also why you only lightly cover the burn area...putting a tight bandage on will only increase the pain when medical professionals have to remove it to further render treatment.

Note medical advice has changed in this matter over the years. When growing up in the 50's I had a red cross first aid book that recommended putting butter directly on a burn area to "sooth" it. By the 70's when i attended multiple first aid training classes, the newer releases of the same red cross first aid book strongly advised AGAINST putting any ointment (petroleum jelly, butter, other) on a burn.

If your burn is just reddening or minor blisters, you may be ok treating it yourself. If you have broken blisters or actually burnt skin, you really need to seek professional treatment as quickly as possible!

I have put this advice to personal use: I suffered small surface 2nd degree burns on my hands while winter camping in the middle of Nevada desert when a lit stove fell and by instinct I reached out to grab it. (Not the best of instinctive reactions!)(I was in the Humboldt Sink, hours from Fallon and the nearest hospital.) Following the advice listed above got me through the process without any downstream complications...I was lucky, but I also knew enough NOT to use "ice" or "salves", and to only loosely cover the area and not tightly bandage it. Admittedly my burns were 2nd degree on only a few square inches of skin on 3 fingers. (It still hurt like hell.)