For what you describe you can't go wrong with a 223. It's a very accurate round, you can buy decent factory ammo, it reloads easily, you can buy fmj rounds that won't tear up the hide, good out to 400 yds or more, short action means a lighter rifle. 204 would be a good choice too, but you won't have as many ammo choices unless you reload.
You didn't give a price range to help with what mfr of rifle. if you're in the 500 range a CZ 527 is hard to beat; light, very accurate, great trigger, comes with rings. You can buy H&R single shot for about 250 (been a while since I priced it). In between is savage, remington, howa, etc. Used you'll pay less, although some (like the cz) are hard to find used.
You didn't mention a scope, but at those ranges (<200 yds) you don't need super expensive optics. Figure on spending 125 up to 250 or more depending what features and brand you want. I've been buying used Leupold vxII's when I can, hard to beat when you can get one for 150 - 175.