The Savage is great. My 10FCP-HS was just about $1100. With hand loads i can get a under .5 average for a day of shooting (not just a few groups here or there). If you go with the savage, get one with the HS stock or Mcmillan (more expensive). As stated, the 5R is great, LTR is great (expensive as well).
Look into the 700 SPS Tactical. It is well under $1000, and will shoot very well. It has a 20" barrel, and Hogue stock. The stock has some sort of bedding, and is usable but you could always upgrade it later. Expect it to shoot .5-.75 out of the box with FGMM.