Can I move away with our son without the father's written permission and how a custody battle works in court?

You need to ask an attorney in IL all of your questions, since that is where the court case will take place.

The analysis must start with the question of whether there is an order of support entered against him, either by a court or an administrative agency, If the answer is no, then where was the child born and is he on the birth certificate? If no court action was filed there is no restriction on you leaving the state. You do not need his permission. If there was a court action filed, then you need the court's permission to leave the state. You can not kidnap your own child if there is no legal restriction on you leaving.

Your question does not indicate how long you were in FL, or how long you and the child have been in IL. The amount of time is important, as it may affect which State has jurisdiction to determine the issues in the event either of you file an action in court. Generally, to file the action in FL, you and the child would need to have been residents of the State for six (6) months or more. If you had been, and have been out of the State of FL for less than six (6) months, it may be possible for you to file an action in FL if you return, as FL might still be considered the home state of the child pursuant to the UCCJEA, an interstate compact of law that applies to situations like yours. Generally, in the absence of a court order or written agreement to the contrary, there is nothing preventing you from returning to FL. However, if you have been in IL for more than the length of time IL law sets as necessary to establish jurisdiction there, you could possible be ordered to return the child to IL if you leave. You should consult with an IL family law attorney to obtain the answers to these questions.