Many many thousands of deer, black bear and moose have been shot by the good old 30-30 Winchester in lever action rifles and many many thousands of others with the venerable 303 British out of army surplus Lee-Enfields.
There are equally many more modern cartridges and ultra-slick plastic fantastic syth-stocked stainless-steel barreled rifles in a whole variety of cartridges. Look at the advice in the above pages and make your choice, but the truth is if it is a centre fire rifle of adequate calibre in a decent rifle and goes bang, then you're set.
I knew an old cowboy who used to kill grizzly bears with his lever action Winchester 35-35. My dad hunted deer for years with a .22 Hornet. I am not advising you to follow suit, I'm just sayin'.
Some no nonsense advice from a sage CGNer.
Rob