Tour

http://www.trivago.co.uk/badaling-50774/palace--castle/great-wall-18065 There are several different parts of the Wall that are open to the public. These are Simatai, Badaling, and Mutieyu. Teach one offers something different - Simatai is the older, wilder, more broken down part of the wall, and you can get some fabulous hikes there without encountering many people. Badaling and Mutieyu are both touristy, but are more convenient and accessible.
 * The Great Wall:**

http://www.trivago.co.uk/beijing-50772/monument--memorial/tian-tan-158134 I think it's one of the most beautiful temples in Beijing. It was built between 1406 and 1420, and among the many things worth seeing inside are The Altar of Heaven, the Imperial Vault of Heaven and The Hall of Prayer for Good Harvests.
 * The Temple of Heaven**

http://www.trivago.co.uk/beijing-50772/boardwalk--promenade--street/tian-an-men-square-16638 You've almost certainly heard of Tiananmen square. It's the largest square in the world - almost 0.8km by 0.5km - and it's just south of the Forbidden City. It was built in 1407, and is known all over the world because of the deadly student riots that occured there in 1989.
 * Tian'an'men Square**

http://www.trivago.co.uk/beijing-50772/museum--exhibition--gallery/forbidden-city-32573 If you want to learn about Chinese history and culture, this is the museum to go to. It was formerly an imperial palace, but it is now a World Heritage site and holds historical elements of the last 500 years of Chinese history. It's very impressive.
 * Forbidden City**

There are of course a lot of other amazing things to see and do in Beijing (and Shanghai too!). You may have noticed that my links were all from the travel site trivago - I'm not trying to advertise for them, I just happen to think they are the best resource for this kind of thing. You should go take a look. Since i don't have any advice for Shanghai, you can look at it yourself: http://www.trivago.co.uk/shanghai-50755