Graytimber -
I'm with Lawdog on this one - a Savage .308 will do you fine.
The one Savage I had was an older 110E in .22-250 that my Dad gave me. Homely as sin, but it would put 4 holes in a 100-yard group you could cover with a dime. The newer Savages are much nicer cosmetically and they have always had a reputation for being good shooters.
As to the .308, it excels as an all-around big game cartridge, although many consider it a bit light for big bears and maybe moose. There is plenty of good, relatively inexpensive milsurp ammo available to practice with. Same can be said for the .30-06 if you want a touch more reach and don't mind a bit more recoil.
FWIW, my favorite rifles are Ruger M77 MKII's. You can pick one up for around $440 for a walnut/blued rifle. You may want to get the trigger touched up but that is an easy thing for a 'smith and when you are done you have an excellent rifle. I've done a number of Ruger triggers myself with excellent results, so it can't be too hard! Every Ruger I've worked with has been an excellent shooter. The Rugers come with Rings and the bases are milled into the receiver, so consider the additional cost of those items when comparing to Savage, Winchester, Remington, etc.