Brian,
Accuracy and effective range in either caliber depends on what targets you have in mind. For varmint shooting (prairie dogs, ground squairrels, etc) maybe 250-300 yards for the .223 and possibly 400-or more for the .22-250. Both cartridges are easily capable of sub-minute-of-angle groups (appx 1 inch at 100 yards).
For larger varmints like fox, coyote, groundhogs, the larger, faster cartridges is favored. Velocities of both cartridges using a 55 grain bullet are around 3200 fps for the .223, and about 3600-3800 fps for the .22-250.
If you plan to hunt larger game (up to small deer) using a .22 centerfire, the .22-250 is a better choice. The .22-50 with fast twist barrel and heavy bullets is adequate on deer. Check your state game laws to make sure it's legal!
An infrequently considered factor is noise and muzzle blast. The .223 is much quieter than the .22-250. For frequent shots at medium range small varmits, the smaller round is better.
Any of the name-brand bolt action rifles are good. Remmington, Winchester, Ruger, Marlin, and Savage. Cheaper usually buys you fewer frills and niceties, but you can still get excellent accuracy from an economy-grade rifle.
HTH
|John