What rights do I have with my unborn baby after he/she is born, and how do I go about obtaining these rights? The mother and I are both young and do not know each other very well, but both of us agreed to have and keep the baby. What are the typical visitation rights California once paternity is established? Can I legally have my name on the birth certificate?

Yes you sign the birth certificate and the mother names the baby with your last name. You shouldn't need any DNA test if you are not contesting that you are or may not be the father.
You can go to court and file for joint custody, visitation and paying child support. The Judge will tell you what you have to pay in support and you will get a court order establishing things like when your visits are. Usually weekends or every other weekend, some holidays and some vacation time in the summer. She will have to inform you of dr visits, school things and any major events in the child's life.

If the mother agrees you can have your name on th birth certificate, ** and I guess you knew her pretty good to be having a baby,

You have no rights till the baby is born. After that you and/or her needs to file for custody of the child.

What custody you get depends on what you and mother can agree to. It can be joint, you each have child 50/50, or anything you both agree to