The rifle you start with should be the rifle you cherish for the rest of your life. It should not be one of the bargain basement rifles that are made by those companies that appear to be in a race to the bottom, neither does it need to cost $2K, although it can if you have the means. The rifle should be one that really grabs your imagination, and that you imagine as the one rifle that is appropriate for all of your future adventures. Thus your quest is for the illusive do it all rifle. When I was a youngster the answer to the question was a Winchester M-70 in .30/06, and today that remains a viable choice.
The important elements of the big game rifle are weight, balance, and overall length the combination of which could be termed handiness. The rifle should not be so heavy as to cause undue fatigue after several hours on the trail. Neither though should it be so light that recoil is unnecessarily punishing. Mountain hunting is arduous, as is hiking in loose sand, and if either of those situations are likely to be in your future, a 6 pound rifle is preferable to an 8 pound rifle, and recoil be damned. There is no benefit to a heavy contour barrel on a general purpose big game rifle. It adds unnecessary weight, and unless very short, results in a poorly balanced rifle. A light contoured barrel is not necessarily less accurate for the first shot; what the heavy barrel allows is fire long shot strings without the barrel heating and shifting the point of impact. Rifle accuracy provides confidence to the hunter, and while the rifle should shoot tighter than we can hold, there is little point in expecting it to stay tighter than a minute of angle. Shooting from field positions, under real world conditions, 2 MOA is much better than most of us can hold, and represents a very high standard which we should to aspire to. If you can shoot two quick shots from a position other than prone, and hit a 200 yard target within an inch either side of your aiming point, the accuracy of your rifle will not be what causes you to miss that buck.
A good trigger is the single most important element a rifle can have with respect to field marksmanship. A good trigger can be defined as one which fires the rifle with no perceptible movement with a single stage trigger, or no perceptible movement once the slack of a two stage trigger is taken up. The weight of the trigger should be light enough to fire the rifle without strain or tremor, within the safety parameters determined by how the rifle is used. A target rifle which is only used on the range on nice summer days, is much different than the big game rifle that is used in the field, in wet numbing cold that makes it difficult to feel contact with the trigger, combined with the excitement of sighting on a live target. This is not the place for a 2 ounce trigger, and even a 3 pound trigger might be too light for those excitable types.
While reliable cycling speaks for itself, you want to be able to run the bolt effortlessly with the rifle at your shoulder. Most modern bolt actions #### on opening, and the effort to lift the bolt, which cocks the bolt and partially withdraws the fired cartridge in a single motion, should not be enough to cause the rifle to twist in your grip. The ejection of the cartridge, fired or unfired, must be positive, ensuring the brass fully clears the rifle. The top cartridge in the magazine should be picked up by the bolt without a hiccup, and chamber effortlessly. The action needs to be robust, as positive cycling, is accomplished through vigorous cycling of the bolt. The b otgl handle should not peal off in your hand.
If you short stroke the rifle, usually the result of gently cycling the bolt, and causes a jam with one one round in the chamber and another in front of the bolt, you should be able to clear the jam without having to take the rifle apart. Thus either a hinged floor-plate or a detachable box magazine (DBM) is preferable to a blind magazine. However, a hinged floor-plate must be able to withstand normal handling and recoil without unlatching and dumping its ammo on the ground, and the detachable magazine should not have the magazine release situated in a place that it can inadvertently be released and the magazine lost. I prefer the hinged floor-plate myself, because it stays with the rifle, but hunters, who in the course of a day might travel from spot to spot by motor vehicle, might prefer the convenience of a DBM.
Sights that you can see under all daylight conditions, regardless of background are desirable. When you shoulder the rifle, the sights should be in your field of view, the instant your cheek hits he comb. Although the use of optics are almost universal, your rifle should be equipped with iron sights, and you should use them enough to ensure competence and confidence, and to understand your limitations with them. The big game rifle benefits from the use of quick detachable scope mounts, so if the scope is damaged you can quickly remove it and use the irons, or mount that second scope in your pack, without the need of tools. Its important not to over-scope your rifle. The primary benefit of the scope is that it places both the aiming point, that is the reticle, on the same focal plane as the target, thus both are in focus at the same time, where as iron sights can never be in focus with one another and neither can be in focus when the target is in focus. Magnification should be sufficient to see your target, and by that I mean the entire target. If you aim at a moose, and all you can see is eye ball, you have too much scope. Magnification should never be at the cost of field of view. On a rifle that fits you, the scope's ocular should not extend much behind the cocking piece of the bolt, and certainly not behind the rear bow of the trigger guard. Shooting at a steep angle from a rested position will bring the ocular closer to you, and if thew scope is mounted to far to the rear, sooner or later you'll get cut.
The stock should fit you; a rifle that fits is handier to use, its easier to shoot well quickly, and felt recoil is significantly lessened by a rifle that is not too long for the shooter. Most factory rifle stocks are too long, and despite the popularity of plastic and fiberglass stocks, a wood stock is easier to shorten and fit a recoil pad to. Its not a bad idea to have a recoil pad even on rifles chambered for mild mannered cartridges, as the recoil pad is less prone to slipping on your shoulder than is a plastic or metal butt-plate. Your trigger finger should easily reach the face of the trigger when the butt is placed in the crook of your arm and your arm held at 90 degrees to your shoulder. You should consider the measurement when you are dressed in your typical hunting clothing. A bulky coat reduces your LOP compared to wearing a T-shirt.
The choice of cartridge is less important than the choice of rifle, with one proviso; if you have little centerfire rifle experience, your cartridge should not be one that ends with the word Magnum. Additionally, a bore diameter of 6.5 mm (.264") should probably be the smallest considered. This is obviously not what is recommended by the minimalists who might well suggest a .223, a .22-250, or a .243, but big game hunting is not varminting, and IMHO, the big game rifle cartridge should be available with at least 130 gr bullets. One might point out that the difference in bore size between the .257s and the 6.5s is a mere 7 thousandths of an inch, which should be inconsequential if both bullets arfe of equal weight and were to impact at the same velocity. But available bullet weights for the 6.5 run up to 160 grs, while those for the .257s stop at 120, making the 6.5s the better choice for a general purpose big game cartridge. Unless your hunting will be restricted to deer only, in which case the quarter bores are an excellent choice, the .257s should be passed over for.