I'm in Cheyoming. Harvest 2-3 Speed Sheep every year. I have also guided many goat hunts for an outfitter.
Let me throw my my .02.
First of all, where to go. If I were to suggest just one "unit", or as we call them, areas, I would have to say area 47. An area south of Casper called the Shirley Basin. Reasons: Lots of game. (there are more antelope in Wyoming than people! True!) Potential for decent size. They issued 1000 licenses last year which increases your chances of drawing. Close to I-80 (you'd be coming from Illinois). Lots of public land (BLM). Wyoming has more public land thatn private! BLM, State, National Forrest, etc.
When you apply in Wyoming, the application gives you three choices. In other words, you can list 3 areas. If you don't draw the first one, the computer automatically enters you in the second, and third if necessary.
Let me talk a little about hunting these unique critters. Flat shooting is good! I've used 25-06, 7mm Mag, and .300 Win Mag. #00 - 400 yards is not unusual. Fun too! Practice though. Practice, practice, practice. Antelope are a thin skinned and "soft boned" animal and go down fairly easy. Once down, field dress and skin & wash as soon as you can! Cool down the meat too. It is not a meat that "rides on the hood" very well. If taken care of immediately, some of the tastiest steaks you'll ever eat.
One fellow mentioned "walk-in" areas. These are listed in the Wyoming G&F site by County in the Public Access link. They do NOT require permission to hunt. There are public areas call Hunter Management Areas which do require written permission. East though, the application and permission slip can be done right on the G&F web site. All in the Public Access area.
BUT! Like I said, there is a lot of BLM land. A good map is the Wyoming Gazetteer & Atlas by DeLorme. It is topo's of the whole state. There is a whole series of them for many states. They will show you how much public land there is throughout the state. Orange is BLM, blue is State, and green is Nat'l Forrest.
Anything else? Get a good range finder that goes out to 800 yards if you can. No sence in guessing.