Hi can I travel by public transport to Summer Palace then the panda zoo then back to Beijing?

Hi, actually, both the attractions are in Peking. I am not clear about your specific location so I cannot tell you the detailed direction. Anyway, the subway line 4 can take you to Beigongmen Station, which is very close to the North Gate of the Summer Palace.
You may mean the Panda area in the local Zoo. From Summer Palace, you can also take subway line 4 to Peking Zoo MTR Station, which is within walking distance from your destination.