If you are trying to hunt certain animals, particularly when required to kill them just with a Bow and Arrow, there are a few tactics you can try to employ:
The game is very large, meaning there are frequently occasions when animals will 'glitch' or get trapped in an area and not be able to move. If fighting something like the Black Panther, you can try to get it to glitch before shooting it with your arrows - it's a tough fight otherwise.
For more docile prey, try getting them between you and the ocean. Then, fire some shots into the group and they will run away from you, into the sea, and drown themselves!
To get shark skins you potentially have to come face to face with one of nature's most fearsome predators. Below are a few tips for making these encounters as painless as possible:
The safest way is to get a boat or jet ski and take it out into the water. Wait for sharks to come round the boat, then shoot them from your position. You can then go into the water to harvest the skins.
If you feel like being more hands-on, dive off the edge of the boat and let them come at you. They sometimes grab you which triggers as QTE (where you press buttons to complete actions) and these, if done successfully, will end with you being able to skin the shark.
Finally, try using your camera and scanning the ocean with it. You should be able to 'tag' sharks and then head out in a boat to find them - this will make it so you don't have to hang around out at sea waiting for them to come to you.
To dive into the water (rather than just jumping and flailing around), you need to press jump once to leap off wherever you are standing, then press it again in midair. This will execute a flawless dive.
If you are wondering how to dive underwater, simply look down and move in that direction. The water has to be deep enough for you to be able to go underwater.
If you don't feel like you are finding enough goats on Far Cry 3's epic map, here are a couple of pointers. The goats tend to congregate around one area, particularly to the northeast of where you start, roughly X:442, Y:761 are the co-ordinates. Look for a cave that they are often spotted near.
Also, remember that once you climb the radio tower in a particular region, all of the animal types in the area are shown on the map, indicated by white silhouettes.