The Imperial College
The Imperial College is where Chosen scholars (who had already passed the county and provincial examinations) would go to study for the national and palace examinations.
As a teacher living in Beijing, I find China educational history fascinating; and there is no better place to learn about it the Guozijian (Imperial College) and Confucius Temple. The College and Temple are connected, though they have separate museums, and the ticket costs 20RMB. You can easily spend half a day here. The museum in the Guozijian focuses on the history and format of the Imperial Exam system, and has limited English captions. The other museum focuses on Confucius and his life, it has very good English captions for everything.
The Confucius Temple is a sprawling tribute to the Great Teacher (Confucianism is a philosophical school, and not a religion. The Confucius Temple is a place where people came, in the Chinese tradition of ancestor veneration, to offer sacrifices to Confucius). Students at the Guozijian would come here seek the favor of Confucius to give them success in the Imperial Exams.
The complex is located near the Yonghegong Lama Temple, and the two can be easily combined for a nice day-long outing. There are also many restaurants nearby and it is a five minute cab ride (ten by subway) to the hip areas of Houhai and Gulou. The line 2 and line 5 subway lines stop right in front of Yonghegong and from there it is a five minute walk to the Guozijian.