I won't say it is the best budget rifle, but here is mine. It is a New England Firearms Handi Rifle. It is .30-06 and well within your price range. And you can have barrels in other calibers fitted to it. I have only the .30-06 barrel for this rifle but I have another Handi Rifle that I have a .357 magnum barrel and a .50 caliber muzzleloading barrel for it.
I have fired the '06 nine times at deer ranging from 25 to 125 yards and have killed nine deer with it. None required a second shot. Some died in their tracks. Some ran but none more than 100 yards.
With the butt stock shell carrier I can get off a second shot almost as fast as with a bolt action.
My .357/muzzleloader. I have killed two deer with the muzzleloader.
(Luke 11:21 KJV) When a strong man armed keepeth his palace, his goods are in peace:
(Luke 22:36 KJV) Then said he unto them, But now, he that hath a purse, let him take it, and likewise his scrip: and he that hath no sword, let him sell his garment, and buy one.