That sounds right--
I'm in Illinois, too.
As far as I know, you have to have you're current FOID, and for handgun I think minimum age is 21.
Then, since its a private transaction, as was said, just keep a "bill of sale" for both parties with date, gun id, ser #, and names of "from" and "to".
Don't need to file that paperwork with anyone, just keep it to yourself.
It's better that way anyway--fewer people know you have it.
A picture, too, doesn't hurt, in case it ever gets stolen.