Hi. 5N in Beijing is not too much. You can have a many places to go and arrange a relax tour there. I think 5/6N in Shanghai is too much. I think 3D2N is enough. If you wan plan to places around Shanghai, such as Hangzhou, Suzhou or some watertowns, it will bw all right to have more days there. From Beijing to Shanghai, by train is ggod choice, for the bullet trains are fast (ten hours) they are all night train. Ticket fare is 452/542. As the current flight info, air ticket is much expensive. See:
https://www.travelchinaguide.com/china-flights/Flightsearch.asp

It is okay to travel independently, but you need to prepare more info about the travel details by your own. These two cities all have good public facilities. If you can not think it well, consider to take part in a tour, which is easy for and after all this is your first time to China.