I use and heartily recommend a .223 Remington, particularly for a beginning shooter/hunter. A .223 rifle is a real joy to shoot, with low recoil and minimal muzzle blast compared to larger calibers (or even bigger .22 centerfires). Ammunition is inexpensive and easy to locate, and finding an accurate factory load is relatively easy for pretty much any rifle. Pretty much any "regular" factory rifle will get the job done for predators and varmints under most applications. You can get into a "budget" gun for about $250 or so new. I'm talking about the NEF single shot break action with an entry level scope. The sky's the limit from there, but you can set yourself up with a decent bolt action (Savage, Remington, Ruger, Winchester, etc.) and good scope (Leupold, Bushnell Elite, etc.) for about $6-800, depending on the particulars. Of course, there's a lot to be said for the afore-mentioned 12 gauge shotgun with #4 buck. That'll really knock 'em down and has all the range you need if you're calling in the brush. Feel free to drop me an email at dtwiebe@yahoo.com if you want to chat more about the .223 and coyote calling in general.
Good luck with you new pursuit!