Where to find animals ?
Animals spawn mostly in fields except for pigs and rabbits who seem to spawn more frequently in forest. Also, chickens spawn in towns. However, it may be easier to find an animal in a field since the open field is more visible than the forest.
By following some simple DayZ rules, you'll be able to safely hop from field to field and search for cows, lambs, pigs or sheeps. Once you find one of these, be sure to look around and search for potential threat. Since most animals spawn in open fields, players are the biggest threat and they will be able to spot you easily.
Rabbits move the fatest. Rabbits will be extremly difficult to kill and will yield the lowest amount of meat (Watch this for more info). Other animals are mostly static and will be quite easy to get to.
Choosing a weapon
Choosing a weapon for the kill may seem trivial but it can save your life. Remember that sound in this game is really important. Getting heard from far away can attract unwanted attention and can easily get you killed.
The hatchet would be your weapon of choice. Quiet and powerfull, it will be easy to kill, gut, and leave unheard.
The Crossbow will give you about the same result as the hatchet, but at longer range. It may be useful if a zombie is near.
If you don't have any space left to swap in and out your hachet, using a handgun is really your best choice in this case. Try to get out of the field really fast and do your business far away so you don't get caught by unwanted visitors. Most handgun requires two shost to kill an animal, so you may want to conserve ammo.
A silenced weapon would be a nice choice, but I wouldn't waste any ammo on this if another weapon is available.
Last, any primary gun if nothing else is available.
Gutting and results
To gut an animal you need a Hunting Knife.
Cows: 8 pieces of raw meat
Sheep: 5 pieces of raw meat
Pig: 4 pieces of raw meat
Lamb: 3 pieces of raw meat
Rabbit: 1 piece of raw meat
Chicken: 1 piece of raw meat
Cooking meat requires wood and a box of matches. To make a fireplace, you must have wood in your main inventory. Then, right click on the box of matches and choose "Create fireplace".
Once this is done, you can light a fire. From there, it's easy to queue actions. So you light the fire, then cook the meat, then put out the fire and then take the cooked meat from the fireplace. The faster the better since you will lower your chance to attract visitors.
Also worth mentioning is that you can use oil barrel in towns (such as Cherno or Elektro) and light them up. This will also give you the option to cook any meat you have on you.