It all boils down to bullet placement. I have used a 30 calibre weapon in Africa with a 165 grain bullet that has taken everything from elk sized game down to coyote sized game.
The recoil is relatively mild and ammo is less expensive than the whiz-bang magnums.
Which make? Go to a gun store and try different weapons and see which one fits you best as far as price, cheek placement, etc. Be careful though, you may end up with a safe full of weapons.
I agree with the 165 grain bullet, my favorite for .30-06 and .308. I also use them with my .300 Winchester Mag. Better sectional density than the 150 and flatter trajectory than the 180.