For deer, check your local regs from G&F. Some state set minimum caliber for big game (e.g. deer) hunting, usually .243. If whitetails are the primary game you intend to go after, the .243 is an excellent choice. If coyotes are your main quary the 22-250 will work great.
The .22 centerfires are quite capable of killing deer, if you use the right bullet, but stepping up to .243 gives you more margin for error with the shot. I know some on this board use .223 for whitails quite successfully, though I would personally not choose to use that small of a caliber.
I would recommend you go with the .243, and get 70-85 grain varmint bullets for the coyotes, and 90-105 grain big game bullets for deer. If you don't want to have to change scope settings it is possible to get 85-87 grain bullets for both varmints and big game (just make sure you keep them seperate, using varmint bullets on deer will cause horrible surface wounds that will take a long time to kill the deer - probably from infection, and the big game bullets will likely overpenetrate a coyote).