If your looking for the best value like you said, check out the line of Savage rifles. Depending on the chambering you want you can get the models 11, 111, 110, 16, 116 or the American Classic which i believe is the 14 & 114. These are all bolt action rifles and chambered for several different cartridges. And Savage has a reputation for good 'out of the box' accuracy. The money you save on the rifle can be applied to a higher end scope and still be within your budget. The Rems, Brownings, etc are nice but cost a bit more.
I just purchased another Savage (#6) for my grandson. It is the 11FYCAK. I got it in 7-08 and it is the youth model with detachable mag and muzzlebrake. I'm 5'9" and I really like the way this rifle fits me with its shorter stock. He has about 9 more years to go until he starts hunting. I just hope I don't have this rifle wore out by then.