Since you are getting started and have never done this before, I think using your Mosin will be just find until you figure out if you will like this or not. Especially if you shoot your Mosin well, like you mentioned.
You'll have to check GA hunting regs, but definitely for deer and probably for hogs, by law you will not be able to use the cheap FMJ 7.62X54R ammo you have probably been using. You will need to find some hunting ammo made for expansion, and a quick google search will produce some results you can begin considering. Make sure you practice with your chosen hunting ammo, as the point of impact will probably be different than what you have been using.
As for any game animal, you need to put the bullet into a vital area. Brent's suggestions for studying hog anatomy is excellent, and you can google deer and coyote as well. Most folks I know take a little more liberal approach to shot angles on varmints, as meat conservation is not a concern.
Those are some basics for starters. You will also need to study hunting tactics as well as do some homework on the property to understand where the animals bed, feed, drink, move, what times they come out, etc. I'm certain more folks will come along and fill in the details.