Would I be able to move out of state if i was never married to my daughter's father, he is in prison, a multiple felon, and never paid child support? He wants visitation but doesn't want to support her.
You do need to consult a family law attorney -- was there ever an order of paternity and/or a visitation.custody order?
You may need to get the courts permission to move out of state -- which would not be a hard thing to do. Further, even though the father is in prison that does not relieve the obligation to provide child support. Whether he may or may not get visitation depends on what he is in prison for -- there are certain statutory crimes in Illinois that a person cannot get visitation if they have committed and been convicted of those crimes.
Consult an experienced Family law attorney in yoru area -- if you do not know of one the Illinois Bar Association operates a lawyers referral service called IllinoisLawyersFinder. Call them and the attornies on that list will privide a 1/2 hour consultation for no more than $25. You certainly need to know your rights and protect your daughter's rights.