I would get a shotgun. A .22 would be okay but requires control and precision which can be hard to do when you are excited and the adrenaline is pumping. If you miss then the bullet keeps on going which could be several hundred yards. A 12 gauge with buck shot has an effective range of about 75 yards depending on the load. Even a small game load would take care of a coyote at 25 yards or so. If you are still worried about hitting something behind it you can always step down to a 20 gauge. In either case get you a semi auto or pump action. Single shots are literally a pain to shoot.