Whatever you do, I wouldn't go to the Badaling section of the Great Wall. Go to one of the ones that are a bit further out instead e.g. Mutianyu, Simatai, Jinshanling.
I recommend eating cheap street food but I also recommend the Peking duck at Dadong. Fancy but still cheaper than an equivalent meal back home. Might need to book in advance, or get a ticket and wait for 20 minutes or however long it takes.
The boyfriend and I splurged on front-row seats for an acrobatic show. Again, worth it for us - we pretty much never get front-row seats to anything, and it was still within our price range.
Visit the hutongs - they are so charming and full of character.
Forbidden City and the nearby Jingshan Park. Summer Palace, Shichahai, Tiananmen Square, etc. I liked the candy blower on Wangfujing snack street if he's still around. If you have time, the Bell and Drum Towers, if I recall correctly one of them has a really dark, narrow and steep staircase which has kind of an adventurous appeal.