The 10/22 could work but I like a little more power for yotes and the shotgun with some buckshot will work within a certain distance. I like the new Ruger American Predator at its price point for a good rifle, cabelas had them a few weeks ago for $379. I'd go with the .243 (or bigger) it will do everything you need and give you some options if you want to hunt something larger since its an all around good hunting round.
http://www.ruger.com/products/americanRiflePredator/models.html