Good thing to have around
Betadine and triple antibiotic ointment are a must have in your emergency kit (NO peroxide since it slows healing and can even damage tissue)
Vets are not always needed but highly recommended depending on the extent of the wound / burn or problem.
How to know if a vet visit is needed or not in a case of an injury is really a matter of experience and common sense.
Severe burns cannot be treated at home, they need to be assessed and will require anti-biotic treatment to prevent infection. Same will apply with extensive injury.
When in doubt ALWAYS consult a vet.