My sons girlfriend would like me to adopt her baby. What do I need to do for this to happen

You can file to adopt in your county's probate court. So long as the mother consents, and the biological father does not oppose, the court will likely grant the adoption.   As an alternative, you may want to just seek guardianship. It is easier to unwind a guardianship than an adoption and it gives most if not all of the same powers.