That's a very tall order to fill with one rifle. If I had to go back to just one chambering, it'd be the .30-06. I'd consider the 7mm Remington Magnum, but I already have the .30-06 and am comfortable with it.

I've used the .30-06 on coyotes. It's not too great if you plan on selling any hides. 80 rounds out of an '06 over a prairie dog town will make you remember the experience if you're shooting from prone. But it's just fine as pronghorn/deer/elk/caribou/moose rifle and it's also perfectly adequate for black bears, and with a heavy, well-constructed bullet will take down a grizzly.

As far as the 1000-yard "target shooting" goes, the kit I use for that isn't at all at home come hunting season. I use a .308. It weighs 13.7 pounds all on its own. The optic on the lowest power is still too high for the timber I hunt elk in. Long-range hunting/shooting is a game of specialization. What you're asking is for your workaday F-150 to take on Z06 Corvettes at Laguna Seca. Particularly at the price point you specify, you're likely to find yourself disappointed.

Instead, you might focus first on being the best hunter you can be out to 200 yards.