Use the verb pursue when you're chasing after someone or something. A hungry lion might pursue a zebra, the paparazzi might pursue a celebrity, and a young dancer might pursue her dream of becoming a prima ballerina.
Pursue originally meant "to follow with hostile intent," from a Latin root word, prosequi, "follow, accompany, follow after, or follow up." Today, when you pursue someone, it's not necessarily with bad intentions. A police officer, for example, might pursue a criminal, and your dog might pursue your cat around the house just for fun. If you pursue a career in journalism, you are working to achieve it. Likewise, you pursue a strategy or a hobby if you do it consistently.