I'm confused, you say you want a rifle
, but menton $250 rifles you are looking at. What type of hunting do you have in mind?
A 223 is OK on deer sized game at close to moderate range. In the right rifle, and with good ammo it can be a tack driver, but is not suitable to
For general purpose hunting a 243 is a step up and 308 will take the largest game in North America. Ammo for either is readily available in stores and is reasonably priced, although not as cheap as 223. The 260 and 7-08 are in between 243 and 308. Both are good rounds, but not available everywhere and more expensive.
There are lots of choices in rifles, but I need to understand your budget better to give advice. If you want a quality rifle avoid the disposable models such as the Mossberg and Remington 770. I'm undecided about the Axis. I know you can do better for only a slight amount more.