This is certainly not an all inclusive list and I'm sure Trip Adviser or whatever is better but here are some things you may want to check out culturally in the city if you have time.

Wangfujing Snack Street
Market street where you can buy souvenirs and eat bugs on sticks (or just look at them).

The Great Wall
The closest section of the Great Wall is called Mutianyu. It's about a 1.5 hour drive. If you find a driver to hire a reasonable rate should be about 600 RMB for 7-8 hours round trip (that's < $100 USD). That includes 1.5 hours driving each way plus a couple hours to hangout/hike at the Wall if you want. This doesn't include the entrance fee to The Wall. You can also pay extra for the chairlift up and toboggan ride down. Those are all separately purchased.

Mao's Tomb/Tienanmen Square/Forbidden City
These are all right by each other in Beijing and you can get there using the subway or a taxi/DiDi.

Temple of Heaven
You can get there on the subway or a taxi/DiDi.

Lama Temple
You can get there on the subway or a taxi/DiDi.

Summer Palace
You can get there on the subway or a taxi/DiDi. It's a bit further away just on the outside of Beijing. This place is huge so you could walk around for hours if you like.

Silk and Pearl Markets
If you want to buy souvenirs, knockoff things, etc. these are decent options to shop. Be prepared to have to barter, a lot. It's tedious and can take a while to get a decent price. You may have to walk away and come back. I'd say start at 10% of their original asking price and then work up to a final price around 30-50% of the initial asking price. They'll probably laugh and pretend like you're crazy but it's part of the game (I think). It just sort of depends. They will only go so low for foreigners. Or if you don't want to mess with it just overpay and get out of there.