The 6.5 creedmoor or 260 is a great candidate for what you want. It can go as far as you want, it can kill anything you want very efficiently, it is easy on the shoulder, excellent ammo is very available, handloading components are readily available now, and there are growing numbers of manufacturers producing rifles in 6.5 creedmoor. In truth, even out west, the majority of your big game shots won't be over 300 yards, as well as coyotes. Thus, the creedmoor is more than enough. Also, if you need to go 400-700 yards on a shot or want to plink paper out that far or further, it'll do that too as long as you practice.
$2000.00 is plenty for a good set up. (Rifle and scope) and the. Buy yourself a good rangefinder. Leica 1600-b is great, and a few of the bushnells for the price are actually very good. don't cheap out on your scope. If you're gonna walk mountains, don't buy a 12-18lb pig, you'll likely regret it.
Leupold, shv nightforce , trijicon accupoint, upper sightron, Zeiss conquest, upper Burris, mid-upper range vortex, all have excellent scopes in your budget.
There are several good long distance calibers out there, but few, are as gentle, potentially accurate, and capable of penetrating deep as a 6.5 in an all around package.