I like Jay's option here with the Savage .308 tac
I like Playboy's line of thought, but I think .223 might be a little too demanding for a 'new' LR shooter, but I'm biased towards heavier bullets in general. If you do decide to hunt with your LR rig, you have much better options and versatility in the .30/7MM cartridges.
If you stick with Savage, Tikka, Remington, you can (somewhat) inexpensively convert over to a chassis gun down the line (with the Savage in Jay's link you're on the way there with a detachable magazine)
Added advantage of .308 (short action) is if you like what you buy, you could re-barrel to a 6.5 Creedmoor down the road; so I'd lean to the Savage with the AccuTrigger as the best starting point. (realistically though you'll just have an excuse to buy a new rifle).
At the end of the day though, as Playboy alluded to re: wind, it all starts with the shooter.
I started 10 years ago with a REM700 Mountain Rifle in .270, 140gr Accubonds, a Leupold Vari-X III and no range finder out to 600. But I also spent 3 days learning some fundamentals from a pretty good instructor.
If you know your variables (ballistics, optics, range/wind) and have good technique, you don't have to spend a ton to get out to 500 and even a little further, especially if you stick with .223/308; that said, probably the easiest entree from a pure shooting aspect would be the 6.5 Creedmoor