I have not hunted big game in a while, but I used to hunt big game exclusively and all the time. With rifles, I have killed with 22LR (not big game), 6mm Rem, 25-06 Rem, 270 Win, 308 Win, 30-06 Sprg, 300 Win Mag, & 338 Win Mag. I have shot singles, pumps, semis, bolts, and levers. From that short list, I have leaned:
With all people I have hunted with, the ones that can "handle" their rifle well, constantly bring home the meat. By handle I mean handle recoil. I hunted with a guy that shot a .338 for years, took the recoil like a champ, and kills game with it every year. Could I ever do that? Hell no, I just can't take anymore than a 30-06. On the other hand, one of the best hunters I know uses a 25-06 from coyotes to elk, which mos consider extremely light for elk. Every year he gets what he hunts because when he pulls the trigger, his aim is true. Why? He can handle the recoil of a 25-06 easily, so he enjoys practice, which results in him intimately knowing his rifle and "handles" her very well so when the shot on an animal finally comes, he knows right where the bullet goes.
When killing an animal, you MUST have the right shot placement and you MUST have a big enough bullet. Many believe you need either shot placement or a big bullet. Nope, you need both. If you shoot a black bear with a 22LR, you probably, most certainly not consistently, penetrate the bear, even if you hit the kill zone. On the other hand, if you shoot a bear in the ass with a 338 Mag, you are not going to collect a bear either. Hit the animal in the Right spot with the right round, you will have great success.
Personally, I HATE long action rifles. I just cannot stand throwing the bolt a long way compared to throwing the bolt half the distance of a long action. Kind of stupid considering the first shot is the one that counts, but I just hate long action.
For me, Of the different actions available, bolt is by far the best. It is strong, quick, accurate, and easy to assemble and disassemble/clean.
As my big game caliber of choice, I love the .308 Win. I suggest the .308 for most people that ask. The .308 is just under '06 ballistics, knocks the hell out of big game, has low recoil, ammo is widely available, and it is short action.
I love the 30-06 for its reputation, personal experience, versatility, and large lists of loadings, (hell look at my forum name!) but I will soon sell mine due to the fact that I HATE long action rifles. The '06 is an outstanding round, but for me, my love of short actions, the love of my lever .308, and my experience with a .308, a bolt .308 is my next rifle no question.