two pairs will start up a breeding programme yeah. Check the raws of the creature in question to find how long it takes to mature, for most animals its a year i think, though some may be less (notable exceptions are elephants at 10 years and dragons at 1000!)
As soon as you have the pairs the females should get pregnant (breeding by spores), you then need to wait for births to start arriving. Once you got some goat offspring (kids) then you can always butcher the original stock if you need something in a hurry, it'll just delay your program till the next generation matures. Not butchering them is preferable though, from my experience theres no 'down-time' between birth and the ability to become pregnant again, this is hard to prove as pregnancy is not announced but after a few years you start to notice the gestation period (time between pregnancy and birth). This means that often an animal can produce a second litter of offspring before their first reaches maturity
Such programmes have a tendancy to start a little slow, but after maybe two years you will usually have more than enough animals to meet your needs.
I usually only have 2 males maximum, like in the wild you only need 1 and a big herd of females for breeding but its always smart to hang onto a second incase you lose the male. (otherwise you find yourself again waiting for an animal to mature to restart the breeding)