The normal hunting method is basically just find a target and shoot it with your weapon until it dies. Smaller animals like birds, rats, rabbits, skunks, raccoons, possums etc can be taken out with a single arrow or small caliber gunshot. For medium animals, you may consider aiming for the head or neck. For even bigger animals you have to aim for their heart or several headshots.
There are times that your shots will not immediately kill your target but will be enough to mortally wound it. It will become slower and struggle to run until it drops to the ground. When this happens, you can end its misery by using your knife.
However, if you’re after 3-star animals (they carry perfect quality pelts and skins), you’ll have to make several considerations ranging from the type of weapon you’re going to use, what part of the animal to aim for, how to approach it without spooking it to get a perfect shot, etc. Legendary animals are an exception to this rule since you can get their legendary pelts in their pristine condition no matter what weapon you use. As such, it’s recommended to use your most powerful guns and rounds against legendary animals when hunting them.
You can also use your lasso when hunting. Once you capture an animal with your lasso, get off your horse without releasing your lasso then slowly approach the animal. This action will slowly drain your stamina so the more stamina you have, the longer you’ll be able to keep the animal secure. When you’re near enough, press the prompt to finish the animal off with your knife.