PSA for Americans because burn first aid is one of the things i am finicky about.
Proper burn first aid requires that the burn be placed under cool RUNNING water for a minimum of 10 minutes. At that point you may take the burn out of the water and use good judgement to determine if you need to do this further (aka if it still feels heated to the touch/painful)
DO NOT use chilled water. This could cause a shock to the area and, if the water is cold enough, cause a cold burn. For the same reason DO NOT apply ice or a cold pack.
DO NOT apply a compress of any kind, even a cool wet one. This compress will not stay cool and wet for anywhere near as long as you think it does and once it warms and dries it will star adding heat and sticking to the area.
DO NOT apply any kind of adhesive dressing. Anything sticking to the would will cause stress and tearing to the area later if proper care is not and/or cannot be applied upon removal and will also trap heat.
DO NOT submurge/immerse/put in still water. The water will heat up around the burn and prevent proper cooling and, if the burn is severe enough, heat up the water to continue the burning process. If you have no other option but to use still water keep the water agitated by either swirling the water or the affected area depending on what is easier on the patient.
You can use burn creams, sprays, and salves just be cautious and they should never be a first port of call as the can chill the area too much and cause a cold burn if used in excess.
I have seen too much improper burn care in American fics and media and half of the American government based burn first aid websites that popped up when i searched had poorly worded (i hope) or straght up wrong care advice.