You should know that you lose say in where the child resides, what kind of education is received, medical care, and religious views. You stopped being the childs parent now.
You are not able to simple give up your rights. It is necessary for someone to make a request that you lose your rights. The state, an individual taking care of the child, then it goes before a judge for a final decision.
If you give up parents' rights, you do not owe child support (unless rights are lost to the Child Protective Services / the state). You also lose visitation and a say in how the child is raised.