Below you'll find a list of all the animals listed in your in-game compendium. There are 178 animals in total, and the trophies/achievements Best in the West, Skin Deep and Zoologist are all related to some degree.
To meet the animal compendium requirements for Best in the West, you merely need to get fifty animals in your compendium. Skin Deep (requires you to skin all skinnable animals) and Zoologist (study all animals) are significantly more time-consuming. To get an animal into your compendium, you need merely study or loot it, which means you should get all the progress you need towards Best in the West while pursuing Skin Deep and Zoologist.
Skin Deep, as mentioned above, requires you to skin every skinnable animal. This isn't quite as bad as it sounds, as a large portion of the animals in your compendium can't actually be skinned - fewer than half can be skinned, in fact. One way to tell whether an animal can be skinned is to simply check the animal's entry in the compendium. If the entry says "Collected" or "Plucked" instead of "Skinned", then it's not required for the Skin Deep trophy. This generally applies to small game (bats, rats, frogs, squirrels, etc) and many domesticated animals.
Zoologist is the most time-consuming of the three trophies/achievements, requiring that you study almost every animal in the compendium (excluding a few missable animals, which thankfully don't count). To study an animal, you need to get close enough to it for the Study prompt (R1/RB) to show up. It's much easier to use binoculars, however, as your scent can otherwise spook animals long before you study them. Mind you, it's not enough to study one variety of songbird, big, dog, snake, etc. You have to study each individual species, and there are quite a few variations... especially among birds.
Some other tips for hunting include:
Corpses will rot if left unattended for long enough. Be sure to loot them quickly else forfeit your spoils.
Animals have variable quality levels, starting either as pristine, good or poor. Dealing damage to the game (generally by hunting it) can degrade this quality, giving you lower-quality pelts/hides/carcasses, etc.
Getting clean kills will allow you to salvage higher quality pelts/hides/carcasses from animals. To do this you'll need to use the correct type of weapons and ammo and score clean kills by targeting vital parts of the animal's anatomy.
Many animals dwell near water. A surprising amount of the compendium can be filled by simply traveling up and down rivers and coastlines.
Some animals are nocturnal and will only come out at night (owls, bats, etc). Others, of course, are diurnal (songbirds). Weather may also effect what animals you encounter. You'll encounter noticeably fewer flying critters when it's raining, for example.
Many animals are scavengers, and will be attracted to food. If you find a rotting corpse during your travels, if you wait within sight you'll probably find some critter crawling around for a bite to eat eventually. This is a fine way to hunt some otherwise elusive animals. You can, of course, also use bait, or just leave the corpse of an animal you killed.
Binoculars are incredibly useful tools for spotting and studying animals at a distance. Just equip them via your item wheel, use them, and aim them around while paying attention to the lower right of the screen. When the options at the bottom of the screen change to indicate options available for spying on an animal you can start fine-tuning your search. Incredibly helpful for finding hard to see or observant animals at a distance, or for studying dangerous animals you may otherwise not want to get close to.
The bow will serve you well, but buying a Varmint Rifle is a fine substitute that has better range, faster fire rate, cheaper ammo and it can still return decent pelts for smaller game.
Your natural inclination to hunt may be to sneak around and search for trails to follow, but honestly, stomping around on your horse can scare game out of hiding. With auto-aim enabled, it can be quite easy to pick off prey while mounted, and is arguably a safer way to hunt some dangerous game like wolves and bears (especially if you keep your distance).
Some animals need to only be studied, killed and skinned to get 100% compendium completion, but others will force you to go through the unnecessary step of tracking them, too.
Near the Elysian Pool area (northwest of the Van Horn Trading Post, southwest of Annesburg) you'll find several new varities of animals; basically "mangy" and "poisonous" variants of other animals found throughout New Hanover. While you can study these diseased animals, they don't show up in your compendium, and you can't loot them, so don't waste your time hunting them.
Most domesticated animals are randomly spawned from a preset pool of possible options. Whenever you find an area where, say, a bull, rooster, or dog spawns, you can save your game, exit to the main menu, reload, then return to that area to possibly reset which exact type of animal has spawned. Dogs seem to be less malleable than bulls or roosters, but the latter two can be respawned until you've located all variants. A good place to do this is a ranch north of Saint Denis.
Notable Hunting Spots
The Heartlands is a large, open area where keeping an eye to the sky is relatively easy. While some songbirds will occasionally fly overhead here, it's a better spot for hunting larger bird. Greater Prairie Chickens, American Crows, Mallard Ducks, Pekin Ducks, Canada Geese, all manner of hawks (Ferruginous, Red-Tailed and Rough-Legged) can all be found here. At night, it's also a good place to hunt Great Horned Owls.
As for ground-dwelling animals, you can also find deer, pronghorns, coyotes, foxes, rabbits and bison in The Heartlands. A herd of bison dwell between the Heartland Oil Fields and the Heartland Overflow area, near the "A" in "New Hanover".
Southwest of the Van Horn Trading Post, in a bend in the Kamassa River you'll find Fort Brennand. While fairly interesting for the loot that can be found there, it's also a great spot to hunt a variety of small game, including rats, squirrels, land snakes, and a large variety of smaller birds, including blue jays, cardinals, cedar waxwings, orioles, robins, songbirds, sparrows and woodpeckers.
Black Bone Forest
Along the western end of West Elizabeth, Black Bone Forest is a noteworthy enough hunting spot on its own, but it's also interesting for adjacent areas that further expand your hunting options. Most importantly you can find the Legendary Buck in this area, which can be hunted for unique materials, which can in turn be used to craft a Buck Antler Trinket which will give you higher quality pelts and carcasses.
In addition to this unique animal you'll find a variety of squirrels, skunks, boars, deer, elk, foxes and cougars. To the east you'll find Mount Shann, where Sierra Nevada Bighorn Sheep and Rams dwell, along with eagles and hawks on various rocky perches. To the south you'll find Owanjila, a lake where aquatic animals like loons, ducks and beavers dwell.
The home of the Legendary Boar, Bluewater Marsh combines two appealing hunting elements; marshland and coastline. You'll find a variety of frogs, toads, water snakes, alligators, boars and snapping turtles due to the former, while the latter makes this an attractive spot to hunt waterfoul, like loons, cormorants, cranes, ducks and herons. The corpses of other game may also lure scavenging birds like hawks, crows, ravens and perhaps the odd eagle or vulture. To get all the marsh-dwelling animals, you'll need to turn your gaze further south to the Crawdad Willies, but for a nice mix of animals, the Bluewater Marsh is a good hunting spot.
Old Harry Fen
Southeast of Braithwaite Manor (which itself is south of Rhodes) you'll find a forest rich in game. Here you can cunt whitetail deer, boars, squirrels, rabbits, anf foxes, but the area is also home to the elusive panther. You'll typically find these dangerous cats lurking along the southeastern edge of the forest.
Prime marshland, you'll find this interconnected chain of islands southwest of Saint Denis. You'll find many of the same animals here as in the Bluewater Marsh, including alligators, snapping turtles, boars, frogs and water snakes, but there are also a few critters here you won't find to the north. While the odd badger, fox and rat probably isn't of interest (considering you can find them elsewhere), the crawdads and egrets that dwell here are much more uncommon. You can also find a variety of orchids in this area that aren't present elsewhere.