When you take after somebody, you behave and look like this person.
EG: She takes after her mother not only in looks but also in appearance.
When you take down something, you write it.
EG: The professor asked us to take down notes of what he said because we wouldn’t find it in the book.
When you take off something, you remove it.
EG: After an exhausting day, the only thing he wanted was to take off his shoes and clothes and lie down.
When you take care of something, you look after it.
EG: She took care of my plants and dog while I was away.
When you take up something new, you start a hobby.
EG: She took up photography and she is fascinated about it.
When you turn up, you show up; you arrive at a place.
EG: Everyone was expecting him, but he didn’t turn up.
When something turns out, it results in a specific situation.
EG: He started his own business, but things didn’t turn out to be as successful as he though they would be.
When something turns into something else, it changes; it becomes something different.
EG: Caterpillars turn into butterflies, isn’t that pure magic?
When you turn something down, you refuse it; you decline (usually an offer or invitation).
EG: It was a very promising job offer but he turned it down because he didn’t want to travel that much.