Legal Question in Family Law in California

You need a court order to establish the custody and parenting plan. File and serve a paternity petition with a motion for custody, visitation and support.

Not until the court says you do.

You should bear in mind that a child should have two parents in his/her life. If you want full custody (by that I presume you mean sole legal and physical custody) you need the court to order same. However, you should be aware that you will need very good reason for the court to award sole legal and physical custody. Also, unless the father can be deemed to be unfit, the court will order a parenting plan and expect you to cooperate with the child's right to have his/her father in his/her life.