ORIGINAL: game4lunch
Rather than deal with some of the more obvious aspects (where, when, how), let's touch on something Alsatian mentioned; WHAT CAN GO WRONG.
G.P.: Be aware that you may be required to have completed a hunter safety coarse and have to prove it. You should go to the Colorado, Wyoming, Montana, Idaho Game & Fish websites and familiarize yourself with the hunting rules and regs. Each state does things differently. "I've got blisters on my fingers". (John Lennon)
This is a good point that game4lunch brought up. I also live in Illinois and it is imperative that you have taken a hunter safety course! You must show proof when you purchase your tag, so don't forget to put the card in your wallet. It's a long drive to Colorado to find out when you get there that you don't have your card.
A point comes to mind on wilderness areas if that is your destination. Regulations state "no wheeled vehicles allowed into designated wilderness areas". This includes carts to wheel the meat out. It's either hike in or horseback! If you down an animal, and don't have pack animals then you better be in shape to pack the meat out. Some states, like Wyoming, it is state law for non-residents to have a resident guide you into wilderness areas. You stated that you will use an ATV, so that rules out designated "wilderness" areas. You can narrow down the areas you can hunt by going through a process of elimination like this.