Livestock are only obtainable from a trading post at the beginning, but once obtained, you can start breeding them by splitting your herd into multiple fields.
Livestock are a good source of food, but take a while to start producing. Although chickens produce eggs while their pasture is filling up, they only produce a decent amount of food once their pasture is full, and the farmer starts killing them for food. Cows and sheep reproduce slower, and only return food once their pasture is full, so can take a long time to begin producing food - but they have the additional benefit of producing clothing materials: leather and wool.