The extent of thermal burn coverage and specific sites involved are often at least as important as the degree of burn. The main issues are potential shock, inhalation injuries, dehydration, and infection, with the first two being the most pressing.
Basics are:
1) Separate from the source of heat, with concomitant neck & spine precautions. This may involve removing material from the surface of the skin.
2) Airway, Breathing, Circulation -- basic resuscitation first aid
3) Look for signs of inhalation injury: charcoal-colored nasal discharge or sputum, singed or missing facial or nose hairs, hoarse voice, burn or swelling of the face, and altered mental status. Any of these bump you up in priority, as inhalation injury can be a silent killer. Any of these also earn you supplemental oxygen if available, even if you seem to be breathing fine.
4) Remove watches, bracelets, rings, clothing that binds, anything that can cut off blood supply as tissue swells. Tight things will get tighter and can act as tourniquets to blood supply.
[
Added: This is also where the site of the burn comes in, as circumferential burns around any of the limbs can put one at heightened risk of "compartment syndrome," where the swelling of the tissue itself compresses its own arterial blood supply.]
5) Cool the thermal burn area by temporarily immersing in cool (
not icy) water, as the burn will continue to progress until cooled.
6) Evidence of impending shock earns you elevated limbs and likely IV fluids, if available. Or even just if the burn area is more that 20% of total body surface area, often estimated by
rule of 9s (see illustrations on pages 2 and 3 of this WHO pdf). Any thermal burns totaling >15% of total body surface area can mean enough fluids are lost--because a main function of skin is to prevent loss of fluids--to put one into shock. If it's a risk, often an IV line is put in anyway, as it's much easier to get in before the veins collapse.
7) Pain control
8) Wrap in clean dry sheets and blankets if needed. Monitor carefully for progression of signs and symptoms. The general rule is to "cool the burn but warm the patient."
People used to be wrapped in wet sheets, but that is generally best avoided now for longer term, as it advances the risk of hypothermia. What general timeframe is the story set in? That will affect the standard protocols used.
Added: Shrapnel is another issue, of course.