Many of fox, coyote, bobcats and like sized animals have fallen with a .30 carbine. Deer and hogs have also been taken, but I wouldn't consider it as a big game cartridge. If that's the rifle you want, then certainly get it. Become familiar with it, and spend endless hours at the range. However, I will warn you now....as you get more and more involved in hunting, you will find yourself wanting "big game" guns and other specialty items. You already have a shotgun that will cover rabbits, squirrels, birds, and, if properly outfitted will handle deer, bear and boar. Don't jump in and spend a bunch of money on starting out to hunt. You may not like it at all!
Sidewinder mentioned a Mentor. Your "mentor" should be someone who covers all the things Sidewinder mentioned in his post. I've run into people while out hunting, that to this day, I can't figure out who in their right mind would give them a hunting license, put a gun in their hands (with live ammo to boot), and send them out into the woods to terrorize wildlife and hunters! Have respect for your fellow hunters, wildlife, the land and especially the land owner!
Good Hunting!