To begin the learning curve with a rifle, I recommend the .22 rimfire, preferably a bolt-action in the mid-range of costs. Marlin, Winchester, Remington, Anschutz as examples.
For serious hunting it takes several, given the range of size of animal you point to. Some states, turkeys are legal only with shotguns. It doesn't take much of a rifle to get a clean kill on a turkey, but the problem usually is not ruining the breast meat--and that means a careful shot on the head, usually.
For deer and elk, and black bear, the .243 is approximately the low end and such as the .30-'06 represents the high end. Such as the 7mm Mag and the .300 WinMag extend the clean-kill distance over the '06, but are not needed for shots inside of 300 yards.
Grizzlies? That's getting into thumper country, with such as the .338 WinMag or the .375 H&H.
There are many other equally good cartridges, but the above is a start. You can find a wealth of information about them in Wikipedia.
I'll let others offer "How To" reading.