What is the best way to clean animal bites, dirty wounds, and puncture wounds?

Cleaning wounds: Flushing a newly formed wound with copious amounts of clean water is the most sensible answer. If you had some medical 'grade' Normal Saline that was sterile to flush the wound that would be theoretically a better situation. Do not use toxic substances such as hydrogen peroxide -- these kill good healthy cells in the wound base.

Dilution: An old saying in surgery is: "the solution to pollution is dilution." dirty wounds need to be cleaned. The easiest thing to do is run clean water over them to do so. If the wound remains dirty, then you need to seek medical attention to have it done more completely. Puncture wounds can be difficult as the "pollution" might be deep inside the tissue and not amenable to simple cleaning.