Livestock produce is available via three major options: Harvesting Livestock in the world, sending workers to harvest Livestock from Nodes one has invested Contribution Points into, and raising livestock privately within fences.
Harvesting Public Livestock
This allows one to gather items from mobs roaming world via the player character. One can obtain hides, any meat (except chicken) and milk that way provided one has the correct tools. For milk it is necessary to find Dairy Cows in the world which can be milked via a mini game. There are cows for example in Olvia in a small meadow northwest and there is a Farm southeast of Calpheon.
Using Livestock Nodes
Livestock nodes can be found in many farms all over the world. For this local worker need to be owned and fed. One can obtain chicken meat and eggs from some farm nodes, for example the Bartali Farm in Balenos. Fleece for Knitting Yarn can be obtained from the Lynch Farm high up in the mountains between Balenos and Serendia.
Raising Private Livestock
Raising cattle needs 2 slots within a fence. To start the cattle growth one has to create a Hay Stack from drying 50 weed. The haystack is placed and the cattle appears needing roughly 8 hours to mature. One can afterwards either "harvest" the cattle same as crops or one can try to improve the hay stack quality and quantity via "Breeding".
Chicken feed to raise chicken is made by grinding 100 Barley with 20 rough stone. This feed is placed within the fence the same as the Hay Stack and can be improved via "Breeding".