How would I go about changing my son's last name from mine to his father's?

I don't know that's why I am asking this question.

It depends, is there a father listed on the birth certificate. If no father is listed on the birth certificate then here in Florida the parents can sign a form called a DH432 and on this form the father acknowledges paternity and can request that the child's last name be changed to his on this same form. You should contact the Office of Vital Statistics in your state and they will inform you of the form in your state. Or you may contact your local child support office and they could probably provide you with the information.

How would I go about changing my son's last name from mine to his father's?
You may go to your local vital statistics office and they will ask you what the old name was and what you would like the new last name to be. Simply add the fathers name to the birth certificate.