You can use cows milk, from the store, or fresh, or coconut milk, or buttermilk, or cream, or half and half, or yogurt, or .....
If you want to use all milk for your liquid, freeze it before you add your lye. That will keep it from getting so hot that it scalds the milk. stir well cuz you can't see when the lye is dissolved like you can in water.
Or you can mix your lye into water--use at least the same amount of water as lye. Then you can use milk as the rest of your liquid. In that case, you don't add it to your lye water. I add the milk to my oils before the mix. Some people add it at trace.
Milk soaps can overheat a bit easier than water soaps, in which case they get a tan color. they will be darker than water soaps anyway.
I often use either cream or buttermilk, I think the lather is richer than with water.