What forms do I need and how to file them for a voluntarily termination of parental rights?

You need to sign an affidavit of relinquishment.  There are three types of affidavits that some parents sign.  The first is just a general, "I hereby waive all my parental rights and agree to the termination thereof" type language.   Some include this language, but provide for a cooling off period, providing that a parent can file another affidavit in 30 days revoking the termination.  The third type of affidavit not only includes the termination language, but also includes a waiver or citation and a waiver of any future court notices.  With this type of waiver and affidavit, the other parent can proceed through the court system to terminate your rights without letting you know anything else about the suit to terminate.

Signing the affidavit is just the first step.  The next step is for you, the other parent, or some government agency to file a motion to terminate your rights based on the affidavit.  The court may accept or reject the petition.  Some parents attempt to terminate their rights in order to avoid a child support obligation and the courts will sometimes refuse to grant these petitions.  If the petition is because the parent wants someone else to adopt their child so their child can have a better life, they tend to approve the requests for termination.  The purpose of the affidavit is to tell the court that you want to termination.  The purpose of the petition or motion to terminate is to give the judge a good reason why they should.

Any family law attorney should be able to help you and/or the other parent facilitate the termination of parental rights.