The correct answers are I'm going to get a haircut. and I need to pick up my clothes from the dry cleaner.

Let's look at why the other options don't work:

• "I'm going to cut my hair": This means you are going to take scissors and cut your own hair! If a professional is doing it, we use the phrase get a haircut (or the causative form have my hair cut).

• "Take up": This is the wrong phrasal verb. When you collect something that is ready, you pick (it) up.