There are some pretty tight shooting semis out. Used to be if your were going for accuracy... you really needed a good bolt, but those days are changing. The issue is that you will have to pay for it. DPMS... sweet 16 is pretty affordable and a good shooter with cheap ammo. If you are looking for inexpensive though... the savage hog hunter 223 foots the bill pretty well. Comes in around $400 or so, has a semi contured heavy barrel, threaded, and mine shoots fiocchi match at .3x over and over again. I was stunned. If you're just shooting yotes and chucks, 223 is about as inexpensive as it gets for ammo, and its available everywhere.
Your broad question could lead to a lot of good answers. IF bolt... a 22-250 can drive tacks. Little more for ammo... but lot more velocity driving it also. Some might suggest a 243... another good round, which would take you up to medium sized deer if you ever decided to go that route.
Whatever you decide to get, I'd just be sure that you can easily get ammo for it. Nothing worse than having a gun that you want to use, but can't, because you can't find cartridges. (why I stayed with fairly common rounds for your answer).
Good luck and have fun!!