The mother is planning on moving from Illinois to Oklahoma.

The unborn child expected due date is May 1st 2015. The mother and father are not married. How do I establish his rights as a father and mine as grandmother?

Did you mean support and visitation? That's a matter for the courts wherever she may be. He has to establish his own rights, but I doubt keeping her where she is, is one of them.

Your son will have to register on the putative father registry to protect his rights from adoption in Illinois. https://www.putativefather.org/index.aspx

Until the child is born, and a paternity test is done, your son is only the suspected father.

You can't really do anything.

HE has to do this--he has to register on the Putative Father's Registry for both states.

He has no rights until the child is born, though. The Putative Father Registry, however, should establish rights almost immediately after birth--at at the very least, prevent the mother from making a decision about adoption without the father's consent.

If you are hoping to keep her from moving, it won't work.

Once the baby is born, OP's son should file for visitation &/ or custody. If he is then determined to be the father, he will also be required to pay child support (if she is awarded primary physical custody).

Some states also have laws giving grandparents certain rights (subject to primary rights with the parents of the child).

If the birth occurs in Oklahoma, jurisdiction will be in that state, according to the UCCJEA. In other words, OP and her son (the putative father) would have to file in Oklahoma.

Also , where there are grandparent rights, they generally require that a significant bond exist between the grandparents and the child. Such that it would cause harm to the child to not continue the relationship. Obviously that doesn't exist here.

You (actually your son) REALLY needs to consult a local Family Law attorney. A local attorney will be able to advise both of you what rights you have and how to try and enforce them.