grammar

Care about has one meaning: to have it be important to you. If you care about a cause, fashion, your family, etc., it means that those things are important to you.

Care for has a wider variety of meanings. It can mean something similar to care about, but it's often used to talk about people you care about. For instance, you care about the environment and you care about your family, but you also care for your family because they're important to you and you love them. Care for can also mean "take care of," as in "I'm caring for my aunt's dogs while she's out of town."

Not care for, in addition to meaning the opposites of the meanings of care for, can also mean "mildly dislike," as in "I don't care for the color purple."