I'd stay away from the Remington 770. I don't like the action. I shoot a Remington 700 and I'm pleased with it. The best out of the box accuracy I have had with any rifle was the Savage 110. You might want to consider an 111. Have you thought about the Marlin XL7 or the Stevens? These are great entry level rifles. The Marlin is going for $250 and you can use the money you save on quality optics.

Speaking of optics, buy a quality scope. Too often I have seen guys buy a top notch rifle & complain they can't hit anything with it. closer examination usually reveals a $30 scope on a $1,000 rifle. A blurry scope or one that doesn't hold its zero (very common) is worthless