The Great Wall is the obvious day trip from Beijing, and the choice of section matters. Mutianyu (70 km northeast) is the most practical restored section. Simatai and Jinshanling (130 km northeast) are wilder and better for serious hikers; the three-hour walk between them is the best Wall experience available from the city. The Ming Tombs are in the same northern corridor as Mutianyu and worth combining.
The Old Summer Palace (Yuanmingyuan) is reachable by subway and can be combined with the Summer Palace in the same northwest corner of the city.