Here's my take on what you're looking for:
1. Hunting rifle. Something you can pack around going after big game. I'd say something under 10 pounds, including the scope, right?
2. Cartridge. Something that can knock down moose or elk at what a lot of people consider long range, which is ~400 yards.
3. Recoil. Something that will get the job done but won't scare you each time you pull the trigger, right?
4. Price. Everyone has a budget, you just don't want to blow the year's budget on a gun that can out-shoot you, right?
I'll start with suggested cartridges and why.
I'd stick with standard cartridges so you can buy ammo at the nearest store. Wildcats can be your doom if you get in a pinch for ammo. I'd steer clear of the "super magnums". The Remington Ultra Mag and the Weatherby Magnums don't give you all that much flatter shooter of a gun, but the recoil will pummel you, and the ammo isn't cheap. The older standard cartridges will cost you less to shoot, will perform equally well (as far as you'll be able to tell) and will hurt less when you pull the trigger.
.300 Winchester Magnum. An established all-around cartridge, good for pretty much anything you're likely to run into. If you're likely to keep your range down under 250? yards, I'd say to go with the venerable 30-06. Think of the .300 WM as a souped-up 30-06 and you get the idea.
I'd also look at the .338 Winchester Magnum. Pretty much the same recoil, but a bunch more umph at the receiving end. Will perform better on those large bulls.
I have a friend and a brother-in-law that use a .270 WM on elk. You have to be picky on what ammo you use, and be a good shot to hit the boiler room dead on, but it works. I have shot one as well on a cow elk. Did the job, but not much to spare. :-( The 7mm Remington Magnum is basically a souped-up .270 WM. Better bullet selection. Again, I'd consider it marginal on the largest bulls, but many people use them. A 175 grain Nosler Partition will handily go diagonally through an elk.
As far as brand of rifle, I'm partial to Remington for various reasons. Get a Sendero if you do, it'll be easier on the shoulder and probably more accurate than you. Many other people swear by Savages as a very good gun for the moey. Not many people recommend Ruger for accuracy. Winchester is just getting it's act together after severely burning the gun community with VERY poor quality in the past. Personally I'd look at Remington, Savage, Sako, Tika, and Browning.
Mark in Utah