Several questions: How do you get the cat?

Try raising your cow's affection for you by talking to it, cleaning it, keeping it inside at night and when it's raining. Also make sure you have enough food for it. Theres also a chance it may be sick so if it is go buy some medication for it. Try buying a new milker at Van's shop since thats also a possable reason your cow won't give you milk. For you to get a cat you need to be friend with Romana during the second chapter. If your friends with her during the fall she should give you the cat.

Become friends with Romana to get the cat.
A cow will not give you milk until she has had a baby. If you just bought her, she is still young. Wait until she is mature then buy miracle potion to make her pregnant. And I'm not sure you can get a new milker :/ good thing you can milk them by hand!

Your cow has to give birth first. You get the cat in either chapter 2 or 3. And you can get a cow milker from either Van or by ordering it

You don't need A milker to milk the cow or you can order one, to get the cat just wait a little while and one day auntie romana comes your house and says she found a stray cat and asks you if you want it, and your cow has to give birth to produce milk.

The game gives you a milker at the start does no one else know that? It's on the tool shelf.

You don't need to be friends with Romana to get the cat. You just have to wake up after your wife in Chapter 2 Fall, (year 1 or 2, doesn't matter) and she'll knock on your door and give it to you. For milking cows, you have to wait a year for it to mature, and use miracle potion. After it gives birth (quite awhile after using the potion) it will give you Mother's milk, then one year after giving birth, it will stop giving milk and you need to repeat the process. I'm not sure about the milker, I never use mine. But you might be able to buy another one?..