chase after (someone or something)

1. To pursue someone or something intently and tenaciously. I've always wanted to be a doctor; I've been chasing after that dream my whole life. The security guards chased after the thief for nearly a mile. Looks like that obnoxious barking dog is chasing after a cat.

2. To pursue or court someone romantically or sexually. Mary has been chasing after Joe ever since she developed a crush on him in high school. My brother always spends his weekends chasing after girls in bars around town. A: "Oh, he tried to chase after me in college, but I was not interested!" B: "Which is why we didn't start dating till grad school."