If you buy a vehicle from a private person get the title signed and get a signed bill of sale. Have a copy of your insurance for your car but it might be best to get an insurance binder from your insurance company for the vehicle your buying. The binder will be proof you have insurance for a new vehicle. You should be able to go to the Illinois Secretary of State driver’s license facility and get a temporary drive away permit. The permits used to be for 72 hours but I’ve been retired several years so things could have changed. The Drive Away Permit is temporary registration and should be valid in all 50 states.
The Illinois Secretary of State Police have an Office in Mt Vernon Illinois. You could call there and get direct answers to your questions. The Secretary of State has a web sight with FAQ. If your going to the SOS Drivers Facility in a large Metro area they can be pretty busy and you have to take a number and wait your turn.
Many times it is less hassle to just haul the purchased vehicle on a trailer as there is no registration required if the vehicle is not on the ground. If you decide to do the Temporary permit it might make it easier for the seller to drive the vehicle to the SOS drivers facility, do the transfer there, then get the Drive Away Permit. The SOS say you have 24 hours or next business day to apply for title. The letter of the law says for a vehicle to be operated on a public road the vehicle has to have valid registration, Temporary or an applied for title/registration and insurance. If you have no registration or insurance the vehicle can be towed and impounded (big hassle & expensive).
The reason I suggested calling SOS Police in Mt Vernon is it is a smaller office and more likely to talk to a Police Officer. Here is the listed phone numer and address for FAQ at Secretary of State web site.
SOS Police, Mt Vernon, Illinois (618) 244-8343
http://www.cyberdriveillinois.com/departments/vehicles/faq.html
I just checked and a non resident Drive-Away Temporary registration is valid for ten days.
ftp://www.ilga.gov/jcar/admincode/092/092010100E04250R.html
If you come to Illinois to get a used vehicle make sure you can get the Illinois Title. Many times a person might have the vehicle financed through a non local banking institution. It can take a couple of weeks to get the title from a bank once the Lien or vehicle is paid off. It is possible the SOS will issue a Drive-Away permit on a bill of sale but I’d check to make sure. If you get the title and a Drive-Away permit you take the car and Illinois title back to Florida to the FL license authority, surrender Illinois title to Florida and apply for Florida Title and registration/license. Many times a Police Officer or DMV person has to verify the vehicle and vehicle identification number from a vehicle licensed out of state to be registered in another state.