Acting cautiously means being careful and hesitant, the way you step cautiously into the first dark room of a haunted house.
Whenever you use caution as you do something, you're behaving cautiously. The Latin root, cautio, means "caution, care, or foresight." Thinking through possible dangers of a situation, or considering any problems that might come up, is acting cautiously. You should approach that cow cautiously; she kicked the last person who tried to milk her!