AnthonyC,
You've already decided what to spend, so now the other decisions.
If you want only to hit paper targets and little varmints, then light bullets will work for you. They generally go faster, which means a flatter trajectory.
For whatever it's worth, .224 bullets (the ones for .223 Rem) are available in 36 grain weight up to 90 grains. I'd use the light ones for woodchucks, gophers, prairie dogs, etc. I'd feel OK using the heavy ones, say 70 grains and up, for deer.
By deciding bullet weight, that will help you decide on the matching twist rate (
read this) for the barrel.
Alternatively, buy the gun and match the bullet weights to it. Keep in mind that the twist for heavier bullets will stabilize lighter ones, but maybe not as well
For the amount you've decided to spend, an AR-15 in that price point will be fun and almost certainly more accurate than a Mini14. However, it probably won't be as accurate as the bolt actions folks have suggested. Plus, you can get some decent optics for one of the bolt actions, have a darn nice target/varmint rifle
and stay within your price.
I've shot the Tikka T3, Savage, and Cz527 in .223. All are very nice indeed. This
Rock River is super nice, but above your price point.
At any rate, they're all still better than no .223 rifle at all.
Cheers,
John