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To hunt a hare, just run after it (by pressing 'Q' once if you were walking) and press the spacebar when you're close enough to bite.
To find and hunt elk, follow the scent trail in elk territory (by pressing the 'V' key). Once you find a herd of elk, turn your wolf and if you're close enough, a white arrow will appear over an elk, and that elk's health will be displayed. Real wolves typically go after the weakest prey (why use all the extra energy to chase after something that can run faster and hurt you more?), and if you're hunting by yourself or with just one other wolf (your mate), hunting the weakest prey is your best bet. (But you can definitely try tackling a healthy cow or bull elk if you really want a challenge!) Chase whichever elk you'd like to hunt and hold your spacebar when you're close enough to bite it. Biting and holding prey drains stamina, so be sure to pace yourself! Watch your health as well- those legs can pack a nasty kick! Repeat chasing and biting the elk until either it is dead or you're too low on health to continue.
Moose hunting is similar to elk hunting, except they can be found almost anywhere in the map, not in specified zones. Look at dawn and dusk for them, especially near water.