HI:

I am a solo female traveler who will have 24 hours in Beijing before meeting up with a group. I also will have 24 hours in Chengdu solo.

Any suggestions on things that I MUST see or do?

Thanks.

Can't give advice on this without knowing what the group is seeing. Being solo in either city is irrelevant...

Go to Tea City, if you have ANY interest in good tea. It's not a tourist attraction and there's no way it will be on the tour's itinerary.

Drinking a nice cup of the pu'erh I bought there right now!

The tour is seeing:

Tour Tiananmen Square and the Forbidden City and drive by the spectacular Olympic venues en route to the Summer Palace, where you enjoy a scenic boat ride on Kunming Lake. Walk atop the Great Wall and then explore a Beijing Hutong (traditional neighborhood) by rickshaw.

I am doing Chengdu after the tour and am going to the Panda Research Institute but don't know what else to see.

Thanks.

Chengdu:
River Viewing Park and other parks
Wenshu Temple and other temples
Day trip to the huge Buddha at Leshan
Day trip to Qingchen Shan

Beijing:
Temple of Heaven - I've also visited the Altar of Land and Grain, not especially exciting, and the Museum of Architecture and the altar to Xiannong, very interesting but hard to find
Yonghegong Temple
Museum of Chinese History if it rains
Wangfujing Dajie shopping street and night food market

Have you bought a guidebook?