If you want a belted magnum, I'd look at the .300 Winchester Magnum. The armed forces are now using that as one of their standard sniper calibers, and it can be used for everything from deer hunting to elk, bear, etc.
Lots of rifles available from nearly every rifle manufacturer. I was in Sportsman's Warehouse earlier this week, and noticed .300 Winchester available from all of the commericial ammunition manufacturers in a variety of bullet weights.
Another good choice is the Winchester Short Magnum (WSM) which gives nearly identical performance to the Winchester Magnum in a smaller case. The only problem I've seen is that the ammunition manufacturers are just coming up to speed with providing a wide choice of bullet weights in that caliber.
An extremely good round is the 7mm magnum. It is a .338 necked down to 7mm. Not as many varities of bullets available for it as the .300 hundred, so it is not quite as flexible unless you reload.
Weatherby makes a .300, but then you're buying either Weatherby ammunition or brass and that's extra money.
The key to the whole thing is the riflescope. It is easier to shoot an inexpensive rifle with an expensive scope than an expensive rifle with a cheap scope.
Scopes need to be defined by what you want to do with the rifle. Hunt or target shooting - or a mixture of both. What is the anticipated nearest target and what is the anticipated furthest target? Target or sniper scopes are larger and heavier as the scope will have large objectives for better performance in low light and large turrets to rapidly dial in windage and elevation. Hunting scopes are smaller so they can be easily carried.
For the money, and general use, it would be difficult to surpass the Leica 3.5 - 14. Leica glass is every bit as good as Zeiss Hensoldt and Schmidt & Bender, and the Leica mechanical construction is robust. At $1500 that would give you $1500 for the rifle - and you should be able to get a really good bolt action for $1500.
You have to consider that you can upgrade a rifle. Bed the action and barrel if needed, put in a new trigger, change the stock, change the barrel. You cannot upgrade a scope. It is what it is going to be for its entire life the day you buy it. The only way you improve the scope is by buying a different scope.
A Savage Long Range Hunter with the adjustable cheek weld, muzzle brake, and AccuTrigger in .300 Winchester Magnum or .300 WSM (around $800 new) and a Leica 3.5 - 14 scope ($1450) and you have $800 left for ammunition - now that, would be really hard to beat for nearly any type of use.
