If you're going to step up to hunt, I'd go with a 308. 223 is fine, but for anything larger than a coyote, you're stretching what the bullet will kill. It's not like war and hitting a person. 223 can kill something like a hog, but it's not the best for the job because it doesn't have the knock down power. And when you're talking bolt action without the ability of a semiauto to drop a ton of lead downrange to make up for mistakes, misses, etc., the larger round will get the job done better.
I've hit 300+ lb hogs with my 308 off my target and it's still dropped them enough to give me a second kill shot without rushing. I've also shot a hog with my AR, and I never found it because it took off running like a scalded ape.
308 is a great all purpose round, whether it's for pushing paper, long distance shooting (500yds to 800 without ridiculous optics), and hunting. There are better calibers for certain applications, certainly, but for my money, between the two I'd get a 308. You can always build AR later. AR308s are a different, much more expensive matter altogether.