For 300-400 yards, .223 is a great choice. It will be more economical and have less recoil than .308. I'd also suggest an AR over a bolt action for your intended purposes. The AR will offer features, options and user customization levels that are unheard of compared to bolt actions, even the most common ones like the Remington or Savage. You could also build it yourself, have exactly what you want and be way, way under your $1000 limit.
However, if bolt action is it and that is what you want, you could do very well with a Savage in a .308 with and Accutrigger and Accustock.
http://www.budsgunshop.com/catalog/.../14431/Savage+17826+11111+FCNS+4+1+308WIN+22"
Once again, everything is a trade off. You can get a rifle that can far outpace either the 223 or the 308, but it won't be doing it for less money or recoil. The biggest thing when it comes to getting good at shooting is focusing on yourself until you look around and have trouble finding people better than yourself.
308 is a very popular caliber among hobby shooters and non-pure accuracy competitions because of how many factory ammunition options there are. For many calibers, anything other that hunting ammunition (or FMJ) gets very expensive.