You are free to move since currently you are the de facto custodial parent. I would advise that as soon as you establish residency in California, file for child support to lock in venue.
This situation is not so simple. Per Fisher v. Waldrop (Illinois Supreme Court), even in a case with unmarried parents, you must have court approval to leave the State of Illinois with a minor child. Since you signed a Voluntary Acknowledgement of Paternity, you and the father are legally bound as the parents of the child essentially forever. If you moved out of state with the child and resided in CA for over six months, you might be able to establish CA as the child's "home state" for the purposes of the Uniform Child Custody Jurisdiction and Enforcement Act - keeping the case there if you moved with the child and the father took no action for several months. Just don't count on it. If you move now with the child, and father takes prompt action, he can have an IL court order you to return to IL with the child. Further the Court will not be happy about you leaving the state without court permission or the consent of the father.
As the father, he has rights to visitation, etc for the child. He will also have obligations such as payment of child support, contribution to your prenatal care expenses, medical care cost contribution for the child, et cetera.
I urge you to speak with an Illinois divorce and family law lawyer before you make a move.
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