First of all it is huge, felt like the city is a half of Czechia size.
Forbidden city (1) is the most popular place (you cluld see in all Chinese doramas or movies about royalties) and must see.
Hotels are cheap, rooms are small but facilities and breakfasts are luxurius for the price. We had this infinity pool with downtown view for 80$ (2).
Park of the temple of the Sun is must vist. Authentic atmosphere, older people dancing, a lot of engaging outdoor sports facilities and a mix with traditional culture (3)
Old town is a place to visit in the night to enjoy the lights. Anyway it is too hot in the day. They sell fried scorpions and other cockroaches as a tourist trap. Local people don't eat that. (But salt dried chiken toes is a traditional beer snack you can get everywhere (4)
Business downtown is a masterpiece of architecture and light design. We enjoyed shopping malls as well. They are not Dubai Mall level of luxury, but closest around the world. (5)
Beijing metro was just crazy, but we used it only once on our way to airport. 2-3$ per trip taxi was super convenient and cheap enough not to use metro for trips around the city. (7)
And airport was huge and authentic - stores where designed as traditional buildings but with the touch of modern architecture (6)
P.S. DO NOT eat pizza with Durian fruit in the airport, especially before flight. It was almost fatal experience☠️