Well, lots of must things in Beijing. Just give you a brief idea:
Visit--Great Wall, Forbidden City, Tiananmen Square, Summer Palace, Temple of Heaven, Qianmen Street, Hutongs and Courtyards, Wangfujing Street. So many, you'd better refer to the listed attraction on this page above.

Eat-- of course the Beijing Duck, and the locals snacks you could find casually on the way.This website also lists some good recommendation at https://www.travelchinaguide.com/cityguides/beijing/dining.htm