Do I have to give a terminated employee their pay check immediately upon termination or do I have so many days to do so?

Illinois law requires that you pay the employee the last paycheck, and any accrued and unused vacation time, no later than what would have been the next pay period for when you would have paid that employee had he/she not been terminated.  It is often a good idea to pay them upon their termination so you do not have to worry about any further contact.  But you are not required to do so.