In a nutshell, you can apply for Florida or Arizona using your training certificate for Illinois. For both, you have to contact the state agencies that handle the licensing and have them send you an application packet (or get one in person from your very astute Illinois FCCL instructor). Both require ink fingerprints taken by a law enforcement agency (unless you apply in person in FL, then you can have electronic done at a police department down there), Make sure you get your instructor's NRA instructor # (not just his/her Illinois FCCL instructor number) on the certificate.
Utah requires training provided by an instructor licensed by the State of Utah. The training can be incorporated into a broader curriculum (such as an Illinois FCCA 16 hour course) but must cover all required topics, including those specific to the State of Utah (such as state laws). No use contacting the state to ask for an application, even if they sent you one or your printed one online, it does you no good if it is not signed and stamped by your Utah instructor.
Pennsylvania used to be quite popular but is less so now as they used to accept renewal applications by mail, but had a change in law which requires applications for new and renewal licenses to be made in person. But if you happen to be making a trip to the east coast, might be worth stopping in to a county sheriff's office.
As for the benefits of each...Illinois (at last count) is recognized by 27 states. Adding a FL non-resident license gives you FL, GA, LA, NM, WY, and ND. AZ does not give you FL, and does not give you any additional states covered by FL, but does cost less. UT is recognized by all states that recognize FL except FL and NM, but UT is also recognized by WA. The trade-off is that UT requires specific training, so time and money up front, but much lower licensing and renewal costs than FL.
Cliff's notes done.
NRA Life Member
NRA Certified Range Safety Officer
NRA Certified Instructor - Basic Pistol, PPIH, PPOH, Rifle, Shotgun, Home Firearm Safety
ISP Approved Firearm Concealed Carry Instructor
Utah CCW Instructor