You must first have a barn built by Gannon on your farm. Once you do, they will become available for purchase for 5,000G each. An adult cow can be milked using a milker every day. You can make yogurt, butter, and cheese with milk and sell them. Brushing cows with the Brush will increase their friendship and their milk quality.
As you increase friendship points with your cow, the quality of milk will go up. If you win the cow contest, they will start to produce s-rank milk, and you'll be able to purchase a Jersey Cow.
If your cows to give birth to a calf, buy the Cow Miracle Potion from Mirabelle for 3,500 G. You also need to have an available Birthing Pen, and the cow must be an adult. Use the Cow Miracle Potion on the cow you want to impregnate, and then remember to put Fodder in the Birthing Pen's Feeder every day inside of the barn. After 21 days, the baby cow will be born. It will take 18 days for the baby cow to mature into an adult cow. Cows have a life span of about 6 years.