STORY OF SEASONS: Pioneers of Olive Town

To first acquire a breeding kit, your Animal Handling skill needs to reach Lv.5. This is raised by interacting with your animals. This can mean: petting them, praising them (surprise face over animal), brushing them (only done when they tell you to with a brush icon over animal), milking (again, milkable livestock will have the icon over them when you can do this), or trimming them (sheep and alpaca have this option instead of milking, a few days to regrow their wool). From what I've seen elsewhere, each of these interactions gives you experience to level up your skill.

Once you've reached the suitable level, you'll be made aware of the presence of breeding kits now available at the animal store through a tooltip. Head into Olive Town and the first building on the left, left counter, will be where you go when it's open. Patricia will offer 3 options: Buy Livestock, Apply Livestock Breeding Kit, Apply Chicken Breeding Kit. Either start raising your barn animals up to 8th generation for 10* produce or choose your chickens.

Farming Tip: Generic animals (Cow, Chicken, Rabbit, Sheep, Alpaca) will require the 7th or 8th generation before they will consistently provide a max-rank produce item. Variants, such as the Selkie, Buffalo, Brown Cow, Black Cow, Gray Rabbit, Brown Alpaca, Suffolk Sheep, or Pink Rabbit, will require a lesser generation of maybe 6th or 7th before they will satisfy the same max-rank availability. Apparently it'll only require eight days after purchasing the breeding kit before the animal is born. From there, a few more days will be required to let the animal mature (a full week for barn animals, perhaps less for chickens).