Rem 700 5r in 308 or 300wm, I own both and they both shoot as good as custom guns, rem700 sendero's are also proven tack drivers. Tons of aftermarket parts. I would throw a scope you hopefully already have on it for now and save a little more money for a good optic, you'll only be left with about 500 for scope after you buy a 5r or sendero both deserve a quality optic. At the minimum I would go with a mark 4 leupold or vortex pst which I did then upgraded to a vortex razor 5x20 on my 308 and night force nxs 5.5x22 on my 300, you will thank yourself later for buying good optics, it sucks to spend 2 grand or more on a scope but you'll only have to one time. Good luck.
I just bought a Savage long range hunter in 6.5-284 Norma. I will say that it took me some tinkering with the rifle and a lot of tinkering with the Berger bullets. The Bergers seem to be super sensitive to seating depth. But i wanted hunting bullets because this IS my long range hunting rifle. For targets though the Amax bullets seem to tune much easier. Anyways. Here's how it shoots 3 shots at 150 yards. For consistent 1000 yard shooting you will most likely need to hand load. It's not hard though.
Gun I got on sale $719
Leupold rings $70
Murphy precision 20 MOA rail $119
Vortex 6.5-20x44PA $279