You can hunt as early as your gang settles in Horsehose Overlook. Hunting can be done primarily using bows and guns though you can use other methods like capturing animals using your lasso then finishing them off with your knife. After killing an animal, you can harvest their parts such as meats, pelts, feathers, fat, glands, and more. Harvested animal parts do not spoil and can be kept in your satchel as space allows.
The meat can be cooked and consumed immediately to restore cores. Cooked meat can be stored as provisions to be used later on. Certain animal parts like animal fats can be used for crafting special burning arrows or explosive ammo though other parts can be sold at a decent price to a fence or butcher. You can also donate these parts to Pearson as camp funds or crafting ingredients. Bringing home animal carcasses and donating them as provisions to your camp will help increase your honor and will allow Pearson to cook stew. Eating this will give you gold cores that will last for a long time.
Hunting is an activity that will also affect the growth of several of your stats. Some Challenges are also tied to your hunting, and it is going to be a requirement if you want to complete the animal compendium since normal animals need to be studied, tracked, killed, and skinned to mark one entry as complete.