Hunter Sense
Use Hunter Sense (Z Button) as much as you can. The Hunter Sense dramatically increases your accuracy and firing power. The Hunter Sense makes it easier to hunt from a distance, which prevents the animal from charging or mauling you.
Stealth
Crouch (pull back on the nunchuck) when aiming at your prey from a safe distance. Crouching while firing gives you higher accuracy and a stealth boost.
Bear Battle
The easiest way to kill the bear is to use the Hunter Sense and target the neck. It should only take a few shots.
Cougar Battle
As the cougar is coming down the mountain, turn on the Hunter Sense. Take out your best revolver and shoot the cougar as it begins to exit the bush.