You’ll find your cow in your gift box, and after placing it on your farm (the location is up to you), you’ll be taken through a short tutorial of the livestock feature, culminating in your selling your first jug of milk for 150 shells.
From that point on, you’ll need to accumulate your own alfalfa by growing it just as you would any other crop. It’s purchased for 100 coins per square, and can be sold for 125 coins once harvested, should you choose to not feed it to your cow. Once you feed alfalfa to your cow (it can hold up to three bunches in its trough at once), you’ll need to wait five minutes for a jug of milk to be produced, at which point you can sell it.
Your cow’s accommodations can be upgraded (click on it with the Inspect tool active) to a nice Tiki Hut for 5,000 coins, and after that, a full Barn for 20,000 coins. These upgrades are dependent upon the number of neighbors you have that play the game (unless you wish to use Facebook Credits to unlock them, for either 10 or 20 Credits, respectively), so I hope you have at least one neighbor who plays the game, or your cow will be left exposed to the elements for the foreseeable future.
So far, the Livestock section of the marketplace only has the Dairy Cow as available to purchase (you can buy additional cows for 1000 coins), but it’s likely the more animals will be added to the game as time goes on. For now, though, make sure to head over to the game’s page sooner, rather than later, to ensure you get your own free cow to use in the game.