I feel your pain; deer hunting in Illinois is about as complicated as filling out a 1040 long form (I primarily hunt public land in north/central Illinois). That said, archery permits are available over the counter and are good for the whole state, no county permits for archery. For gun hunting, there are plenty of county permits left for most counties. There is an online purchasing system at the IDNR website in the deer hunting section. Almost all public sites cannot be hunted with a county permit (though an archery permit is good there as well as private) and have seperate permits -- in the online system, they appear as a seperate choice along with counties. There is another page with permits remaining, so check that before you apply so you know there are permits left for where you are applying.
Preparing for what you will do if you get one is a good idea. Check Youtube, there are several videos about field dressing a deer. You can take the deer to someone to process, but you still have to field dress it yourself.
If you plan on buying a stand, you can get a ladder at Diick's Sporting Goods for under $100 -- pretty much no frills, but they work.
Good Luck.