care for somebody/ something means (not) like (e.g. I don’t care for his behaviour.)
carry on (doing something) or do on (doing something) means continue, continue doing something
carry out an order/ an experiment/ a survey/ an investigation/ a plan etc. (e.g. An investigation into the accident will be carried out as soon as possible)
catch up (with somebody), catch somebody up means move faster that somebody in front of you so that you reach them (e.g. You’ll have to run faster than that if you want to catch up with Tom.)
check in means arrive and register at a hotel (e.g. We will get the hotel keys when we check in.)
check out means leave a hotel (e.g. You have to check out of the hotel before 11:00 AM.)
check out somebody/ something means look at (e.g. Check out the crazy hair on that guy!)
check somebody/ something out means look at carefully, investigate (e.g. The company checks out all new employees.)