I've been doing a good deal of research on this matter and in this price range.
Savage Axis is about that price range or less. It has a heavy trigger, but from what I've read, a good gunsmith can make it into a 3lbs trigger for a couple dollars in parts. It has had very good reviews, out of the box and can be a 200-400 yard rifle. They do sell replacement triggers, but they can run about $100 for a rifle that's maybe $260.
Savage 111 series has the Accu-Trigger and some (all?) have the Accu-Stock. The trigger is well regarded and can help with accuracy. The stock has had mixed reviews, especially if you want to replace the stock. The 110s should have the Accu-trigger, but not accu-stock. The nomenclature for the Savage rifles can be found online, to help you understand the difference between a 111 FN and a 111 FXP. Savage has developed a good reputation for quality and affordability.
My buddy who is a very experienced smith and shooter told me that you can get a good Remington 700 BDL for $400 or so if you don't mind minor factory defects, like stock blemishes and scratches (those are the 700s at Walmart). The 700 is very well regarded, especially for new hunters.
I have not researched other makes/models, so the reason I wouldn't recommend another maker is my own ignorance.
I'm leaning toward the Axis. My buddy can tweak the trigger (yeah, it voids the warranty). In central Penna., I won't be shooting more than 200 yards or so, maybe 300. So, I'll put the money saved into optics and ammo.
I think others have pretty well stated the case for the 30-06, as well.