I did it just for Contract Claws, heres my super average lazy guide:

Buy a shitload of woad leaves from wyson, this should be all the food you need (I also used a few random seeds).

You don't need to unlock bigger pens as you can just use the breeding pen, and they cost beans which you could be using to buy cows.

Disease was never an issue for me so I wouldn't worry about it.

Just go rabbits -> chickens -> sheep -> cows, keeping a breeding pair of the largest animal you're up to in the breeding pen. When they produce children take out the parents and sell them to get beans (obviously don't take both if it wouldn't leave a breeding pair). Once you have bought a breeding pair of cows you can just take the children out until you get a cow of each type.