I have a trio of stainless Ruger Hawkeyes that I think are good steps, as the 223, 6.5 CM, and 338 FED will work for anything that I'm likely to shoot in the lower 48.. I'm sure a 308 would work about as well as the 338 FED with 180 and 200 grain bullets and there are a lot more and less expensive factory ammo options for the 308.

When my Wife's niece was going to BHSU, she hunted coyotes and pdogs with a 223 and deer with a 30-30.