Get a .30-06 from Howa
like this one here.
It's right at the top of your price range, but you are getting a lot of gun with it.
I would plan on getting a glass bed job done on the action, that should improve accuracy on an already good factory action. That would be less than $100.
Get the trigger tuned down to around 3.5# or so. That should be about $40. Any lighter and you won't be able to shoot well when it's cold and you have gloves on. Any heavier and it won't make a good shooting target rifle.
You are going to need good rings, but you don't need super premium rings since you won't be shooting dimes.
Glass? You are probably going to want something in the 3x9 variable range if this is a hunting gun first and a target gun second. Too high of a power narrows your field of view and makes shooting game next to impossible.
Get yourself a real good quality leather sling, you wil thank me for that later when you are humping to your stand or dragging your animal to the road.