To care for means to help or give special attention to something or someone. It implies that you are responsible for that person or thing.
Ex: Since the day my grandmother became ill, I have cared for her. I make her meals, do the laundry, and go grocery shopping for her.
Ex: It is difficult to care for babies. You have to feed them frequently, and you will probably be awake for most of the night.
Care for is also used to express how fond of, or how much you like something. In this case, it is usually used to express a negative feeling.
Ex: When the waitress offered me chocolate brownies, I told her that I did not care for anything with nuts in it. I don't like their flavour.
Ex: When Bill asked Sue to be his girlfriend, she had to say no. Sue did not care for him in that way and wanted to remain friends.
The phrase look after refers to taking care of someone or something. It implies that you are responsible for the person or thing. In this case, it is a synonym for the phrase care of.
Ex: My sister was very upset when her goldfish died. My mother asked me to look after her until she felt better.
Ex: I was looking after my neighbor's dog when it got sick. It was upsetting because the dog was my responsibility while its owner was on vacation.
The phrase look after also means to follow something or someone with your eyes as they move away from you. It implies a sense of longing.
Ex: The young boy looked after his sister as she moved to the door of the bus. He would miss her terribly while she was away.
Ex: The jockey looked after the horse as it trotted towards its stall. It was a beautiful creature, and he wished he could ride it.