Request your advice with the most recommended itinerary to explore best spots in Beijing & Shanghai.

Hi, you do make a good decision to have a trip covering the two cities of - one is full of Chinese culture and the other presents the modern China.
I suggest you stay at Beijing for about 4 days to enjoy the Forbiden City on the first day. After tht, you can go to the Tiananmen Square to take a view at the world largest square and the nearby attractions. On the second day, you can go to visit the Summer Palace to enjoy the ancient Chinese imperial garden. Along the visit, you can aslo visit Yuanmingyuan nearby. After that, Beihai Park can be on your list to experience the natural beauty. On the third day, you can go to climb the Badaling Great Wall, on which you had better spend a whole day. On teh fourth day, you can go to visit the Temple of Heaven as well as the Bird Nest and the water cubic.

Shanghai for about 3 days. you can take the first day on the Bund to have a good visit of the modern Huangpu River. If time is sufficient, the visit to Oriental Pearl Tower is necessary. On the second day, you can go to Yuyuan Garden or Ancient Yi Garden to experience the nature. Then you can go to Nanjing Pedestrian for visiting or shopping. On the third day, I suggest you go to Zhujiajiao Water Town to know mroe about the Chinese traditional water town.
By the way, for such a long-time trip, i advise you join in a tour group. TravelChinaGuide is suggested. If necessary, you can contact them by sending a message to travelchinaguide.com|tours

Have a nice trip!