To 35bore - I am in Springfield and get to St. Louis frequently. Was there for meetings during the NLCS but unfortunately couldn't make a game.
To Phoenix Phil - we just bought a house in Rio Verde last year. I'd like to give the desert quail a go one of these days as well as AZ big game. Any advice?
Sorry to be slow getting back to you. Lots of desert quail just about anywhere you go. The problem is they prefer to run than to flush, you just about have to have a dog. The problem there is you'll spend as much time pulling thorns out of your dog's paws as you will hunting. I love to bird hunt but grew tired of it without a dog and the one and only time we took a friend's dog I went home feeling awful for the poor hound.
Big game? Well that's another question. The Kaibab is famous for mulies but is a very tough draw. If you get drawn however it's mostly a matter of you taking the time to do your scouting.
Antelope are great, but once again you have a tough draw. If you get drawn, again put your time in scouting and you shouldn't have much problem finding them. A number of the private landowners will allow you access to their land for a fee. They tend to limit how many they let on so I'd probably recommend it.
Big horn sheep, if you can get drawn you will be contacted by plenty of outfitters willing to take you. And I'd recommend hiring one, this is a very tough draw and you might as well spend the money to elevate the odds of success. It's very likely a once in a lifetime draw. I believe Craig Boddington applied for 30 some odd years before drawing.
Coues Deer are fairly plentiful and in my opinion are a great hunt. I would avoid any of the October hunts, the weather is just too warm and the little guys will barely be moving during daylight hours. Go with the November or better yet December hunts. The December hunts are rut hunts and are just fantastic when you find the does. The downside is they're also a bit more challenging to draw.
Elk are what AZ is probably best known for. For the most part if you find pine trees, you'll find elk. Any of the units contain great elk, some of them supposedly hold better elk than others and I guess that's true. I hunt in 4B which is "known" for smaller bulls, I'll let you decide why I'm using the quotes on known.