You can adopt the following route:
Shanghai - Shenyang - Qingdao - Hangzhou - Shenzhen
Honestly speaking, these cities are far from each other. If you want to save time and leave more time for your Shenzhen trip, you can fly from Shanghai to Shenyang. The flight takes about 2 hours to get to Shenyang from Shanghai pudong airport. If you choose the train from Shanghai to Shenyang, you will spend more than one day on the trip. From Shenyang to Qingdao, there are five trains within a day. However, the trip needs about 20 hours. If you continue to fly from Shenyang to Qingdao, the trip only costs less than 2 hours. There are no direct trains from Qingdao to Hangzhou. If you want to go to Hangzhou by train, you need transfer to Yantai first and then go to Hangzhou. From Qingdao to Yantai, it takes about 3 hours by train. Then, from Yantai to Hangzhou, it takes about 26 hours. Also you can go to Hangzhou from Qingdao by air, it takes more than 1 hour. Finally, you can take the Train T211 from Hangzhou East Railway Station to Shenzhen. The trip costs about 16 hours. So, in my opinion, if you want to visit these cities in a week, you have to choose flight but not train or bus sometimes. You can make a plan according to your economic conditions. You know, on most occasions, flight tickets are much expensive than train tickets. But your time is very pressing. If you always choose train, you will not have enough time to visit each city. By the way, you can do your best to purchase some discounted flight tickets. Actually, sometimes the discounted flight tickets are not very expensive.

Here is the map of China. You can find these cities in the map. Just for your reference:
https://www.travelchinaguide.com/map/