For hotels, it is suggested that you find one around the Tiananmen Square, Qianmen or close to subway station.

The subway of Beijing is very convenient and cheap enough, so please use it as much as possible.
Lama Temple: could be reached by subway Line 2 and Line 5
Forbidden City&Tiananmen Square: they are located together, which could be reach by many city-buses and subway Line 1
Great Wall: at the Deshenmen bus terminal (reached by subway Line 2), you could catch the bus No.919 to reach Badaling directly
Summer Palace: the subway Line 4 is the best choice, besides of it, lots of city-buses departing from Qianmen could reach the park directly.
Beihai Park: it is just located at the north gate of the Forbidden City. Many city-buses could be used, too!
Purple Bamboo Park: over ten city-buses could be used.

Well, you'd better fix on the hotel and then I could give you the detailed itinerary.