What actions do you have to go by in order to change your last name, legally?

Answer 1

You need to file a petition with the appropriate court, usually the local county branch of your state court in the US.

Most courts have a simple form that you fill out, available at the clerk's office in the courthouse. You fill out the form, pay the fee, and maybe go before the judge who will ask you some questions.

Answer 2

It depends on the state. Please try this link.