You basically have two approaches to getting animal products: hunting and breeding. Hunting is simple: make some leather armor, a crossbow, and some bolts, turn on the Hunting labor for a dwarf, and build a butchery and tannery. He'll go out, find huntable animals, kill them (or at least try to), and haul their corpses back to the butchery, at which point your butcher and tanner will kick in.
Animal husbandry is a bit more complicated, but much safer. Buy a male and female of the same species (either at embark time or from the caravans), and don't confine them to a cage. The female will breed regularly, popping out young animals. You can either butcher them immediately or wait until they grow up; waiting gets you more products (young animals have half the meat and fat as old ones) as well as the possibility of having more breeders. In either case, you can mark animals to be slaughtered from the Animals screen off of the 'z' status page.
Some other things to consider:
* Currently inbreeding isn't a problem, and you only need one male on the map to keep all the females pregnant.
* You can make a pit activity zone over an open space and toss animals into it. This can make a simple way to confine your animals without having to cage them, which stops the females from becoming pregnant.
* Butcheries get cluttered fast. Have haulers ready.
* Fat can be processed into tallow at a kitchen; tallow can be baked into meals.
* Dogs make cheap breeding animals, since they breed in large litters twice a year.