I would suggest you consider spending more on your optics than on your rifle.
For example, you can put together a package like a stainless steel Weatherby Vanguard, with synthetic stock, and a Swarovski AV 3-9x36 scope for around $2,000 (with rings, bases, and sling). The rifle comes in all sorts of chamberings that may meet your needs.
If you don't care for the Vanguard, a Tikka T3 Lite Stainless/Synthetic would up the cost by about $150, or a Ruger 77 Mark II All Weather would up it by about $70.
All three of the rifles mentioned provide you with the advantages of a stainless steel barrel and action and a synthetic stock. From what I've read, the Tikka will likely be the most accurate, and probably the lightest. The Vanguard is quite durable and suitably accurate for most hunters' needs, as is the Ruger; you might want to look at Savage as well. Get your hands on whatever rifles you are considering and let the feel of the gun be your guide.
Without the budget to purchase the best of both, buying a damn good scope will serve you better than buying a high-end rifle. Most guns built today have the potential, out of the box, to shoot better than the average hunter will ever realize. The gun chucks the bullet, the scope is your aiming device. If you can't see what you're trying to aim at, or if your aiming device is not good, you will probably miss, regardless of how well your gun shoots. Besides, a nice scope can stay with you even if you part with your gun in favour of a new rifle.
The Swarovski AV 3-9x36 offers potential weight savings over a 40 or 44mm scope, and the superior glass should compensate for the potential of reduced light gathering from the smaller 36mm objective. But, in the very unlikely event that you require service for the scope, repairs may be difficult and time-consuming compared to a Leupold (in fact there is no doubt of that).
If you aren't hell-bent to spend all of your budget, you will quite likely be very happy with a Leupold VX II (or even VX III) or Bushnell Elite, or a Burris Signature Select scope.
With respect to chamberings, you may have stirred a bit of a hornets' nest with your question. Rather than offer an opinion on a specific calibre or chambering, I'd say that you should avoid big magnums and hard recoiling rounds. The first centre fire rifle I ever shot was a 30-06 and the first one I owned was 7mm Remington Magnum. Both are good, but there are a lot of other good ones as well.
Others may disagree with me, probably most CGNers have more experience than me, read everything they write but don't create a caveat against anything until you've looked at in personally.