The three rifles that every gun owner/hunter should own (in my opinion), is a 22 LR (small caliber), a medium and a large caliber rifle. All rifles should be the same type (bolt action, lever action, semi auto) If possible, every rifle should be the same basic model. This will ensure that the skills used in one rifle will transfer to the others.
The 22 LR should always be in a rifle collection. This is the training rifle. It is inexpensive to shoot and has zero recoil. This translates into hours of training at minimal expense and no fatigue. The 22 LR is the platform to learn shooting fundamentals and the muscle memory needed to shoot and reload without conscious thought.
The medium and large rifles follow the 22 LR. These rifles teach shooting fundamentals with a centerfire rifle, which is heavier and has more recoil.
The medium rifle is a moderately powered rifle. Consider the 223 Rem, 22-250, 243 Win, or 6mm Rem. Advantages of a 223 Rem is cost per shot, which is half that of the other cartridges. The 22-250 is flat shooting. The 243 and 6mm can be used for both small game and for deer depending on bullet weight. All of these rifles would have low recoil.
The large rifle is a true hunting rifle. A 30-06 or 270 would be ideal, but not essential. Any of the 308 family (260, 7mm-08, 308) or 30-06 family (25-06, 270, 280, 30-06) will work. Any of these is effective for most hunting in the US.
They are also ballistically similar for shots up to 250 yards. A magnum rifle also fits into this category, but comes with some extra recoil and extra expense. Any rifle bigger than 30 caliber does not really fit into this category because it is typically not needed for North American hunting and has added expense. If elk and bear are all that you will be hunting, then maybe consider a 338 with the warning that this will have a stronger recoil.