Treatment for minor burns (1st degree) and even some 2nd degree burns at our house are very simple...first run under cool running water, to cool off skin area, then apply silver sol gel, (since this is antibacterial- I feel VERY safe applying it & IT IS EXTREMELY EFFECTIVE in relieving the pain too) and depending on the severity of the burn- we may or may not apply a sterile gauze bandage. That's it.
It is said though that cold yellow mustard can be applied to the skin after cooled off with water and blotted dry with a clean towel (minor burns without blisters.) Another page on mustard use, here.
Determining the severity of a burn is usually by these signs and symptoms:
1st degree burn- the least serious, affects only the outer layer of skin (epidermis), red skin, some swelling, and usually pain
2nd degree burn- this is when the first layer of skin has been burned through and the second layer of the skin (dermis) is also burned, blisters develop, skin is splotchy in appearance, severe pain and swelling
3rd degree burn- the most severe burn, involves all layers of skin (epidermis, dermis, and hypodermis)...can cause permanent tissue damage, areas may be charred black or appear dry and white.
Burns are susceptible to tetanus! So be sure to have your recommended 10 year booster shot as needed. It would be good to keep these shot records in a safe and quickly accessible place such as a wallet or even one's phone.
The Mayo Clinic has a nice break down with more info and conventional treatment of 3rd degree (SEVERE) burns here if you would like to read it.