The AR15 and AR10 rifles, properly built can and do turn in .5 MOA and better groups. Those of us who are hunters generally don't need better than 1 - 2 moa.
That having been said, mid-priced bolt guns like the Savage out of the box have a reputation, but not usually the guarantee, of excellent accuracy.
Returning to your choice -- there's been a lot of advice given between the .260, the 7mm, and the .308 and a mention of the .243 Winchester.
Your best bet is to do the following:
- Recognize that ammunition for the .260 Rem will be slightly more expensive than the other cartridges at about $30/box and up
- If you are serious about 1000 yard shooting,you WILL be using premium ammunition. The differences in price for premium ammunition between the calibers tends to disappear.
- The price of premium ammunition plus your fiscal situation and amount of shooting will take you to a decision to reload or not reload independent of the caliber.
- The .243 Winchester with heavy bullets has some capability against black bear. The .260 and up have more.
- The real difference between calibers for long range shooting is in wind drift. The drop is readily dealt with using turreted scopes, but the wind tends to have a random component that makes precise correction difficult.
You can get a feel for wind drift by going to a trajectory calculator like
http://www.jbmballistics.com/cgi-bin/jbmtraj-5.1.cgi and run high BC bullets at typical factory velocities for each of the cartridges.
Recoil
will be a factor in your shooting enjoyment. Heavy bullets in the .308 give recoil somewhat above that which most shooters find comfortable to shoot. Try using the recoil calculator at
http://shootersnotes.com/articles/taming-recoil/. Use the same weight of rifle throughout the series of calculations for the same bullet weights and velocities you run the trajectory calculations at.
Next, try using Berger ballistic coefficients
http://www.bergerbullets.com/Products/All Bullets.html under the notion that you will reload. You will likely find that the .243 115 gr VLD bullet and .260 140 gr VLD give about the least drift at 1000 yards.
Beating the 140 gr 6.5 mm BC requires that you go to 180 gr bullets in 7mm and 200 gr bullets in .308. Two bad things happen. First the muzzle velocity gets to be rather low, and second, your shoulder can tell the difference!
Hence, the .260 Rem seems to best fit the interests you've indicated, but the .243 works very well too, including a reasonable capability against black bear out to about 300 yards.
This was a long-winded way of saying that you should look at the .243 especially if you don't plan to reload.