I think a 223 Remington would be a good choice. With the 223, you have a good selection of relatively inexpensive ammo available with many different bullet constructions. The 223 is a very accurate round. The 22-250 is overkill, in my opinion, unless you' re planning to take shots over 300 yards.

If you have any plans of getting into handloading in the future, you may want to examine the 222 Remington or 222 Remington Magnum. They are both basically the same thing as the 223 but offer a few accuracy enhancements conducive to the cartridge case design. The only bad thing about them is that ammunition is a bit more difficult to find...