Another vote for the .243. It is great for everything from praire dogs to medium sized game.
It is an outstanding round. Perfect for all kinds of varmints, and well suited for deer with good loads and good shots.
I share the feeling of not using it for "big" deer, though my largest deer ever fell to my .243.
Smack dab in the middle of your spread (.224 to .284) lies the .257 caliber, but I think you'd love the .243 Winchester.