My default answer is one of the new Winchester 70's unless they are out of your budget. They ain't a budget gun, but are reasonably priced for what you get and are proving to be probably the best current production rifle you can get.
I'd also suggest the 300 WSM over the 300 Win mag. The 300 WSM has been around 12 years now and is a solid seller. At one time ammo was harder to find and more expensive, but no longer.
It won't shoot quite as fast as the 300 win mag, about 2% slower, but no game animal will ever notice that difference. But you will notice the reduced recoil. You have the option of about 10% less recoil in a gun of equal weight, or about the same recoil, but in a gun about 1/2 lb lighter if you choose.
It also works better in shorter barrels, and has proven to be more accurate.