My game experiences have been mainly focused on a pc; I have attempted several games on a PlayStation 1 and 2, but those were all so colossally bad I have not attempted them in a while.
PCI have played a few games on a pc, here are some of them:
Hunting Unlimited (Various year versions): Hunting Unlimited is an okay game on the pc. The last version I played was around 2008, but most aspects of the game were very similar in the earlier versions. There are a variety animals, from Africa and North America, the graphics were okay the last version I used. Hunting was done by wandering around until you find a group of animals.
One point of note on this game was that unless you used the biggest gun and shot right between the eyes, an elephant could not be killed unless you hit it between 21 and 23 times with any weapon anywhere on the animal.
Deer Hunter by Atari This is the best hunting game I have played.
There are several types of deer to hunt, not any other types of animals, and hunting can be done by wandering around, or (rarely for a game) from a stand. There are a variety of weapons which work realistically, you can even shoot the flying birds. The shooting is accurate and if you hit a deer in the leg or too far back you will not get it.
I recommend this website: Hagadoo.com, for additional maps which were user created with interesting new places to hunt.
Console (Playstation and Xbox)The main games I attempted on a console had a Cabela's name on top.
Cabela's Big Game Hunter & Dangerous Hunts These two games have had a few versions but the ones I played had poor graphics and awful game play. Once you arrive on a map you look for a glowing red dot and proceeded to follow that dot until you reach the animals, which you spook because you are mindlessly following a dot and don't see the animals until you are right on top of them.
WiiI have not tried a hunting game on a Wii but I suspect that if you are playing on a Wii the parts of hunting we like in real life may be dumbed down too much to bear any relation to hunting.
You may just want to look for a Wii game where you shoot things rather than try to find a hunting game.