The AR is a nice gun, but seems like the lower end ones start at around $800. Im a huge fan of Browning. Never seen one that would'nt shoot MOA or better out of the box with factory ammo.
If you are going to do an AR, Id get a minimum of a 20" barrel for varmint hunting. Bare in mind, you give up approximatly 50 feet per second in velocity per inch of barrel on a .22 caliber barrel. Thats why I dont like 16" for a coyote rifle.
I much prefer a bolt gun personally. Although the ARs are cool, you have to make sure you keep them cleaned and well lubricated because reliability can be an issue especially if you neglect it. For me, If I were to get an AR, Id probably go with a Bushmaster, but Im a far cry from an expert on ARs.
For hunting in the midwest, I like a .223, but out west, Id probably prefer a 22-250 with a 24" or 26" barrel. Thats some go,go.
The new Predator camo rifle from Savage is an awesome tack driver for the money. But in 22-250 its only a 22" barrel. Thats a bit of waist cause with that short barrel you are burning alot of powder out side the barrel once the bullet has left the barrel. In a 22-250, you need a 26" barrel to burn all that powder. Id do a .223 in the Savage Predator. Good chance is will shot well below MOA with a good Factory ammo.