I will be in Shanghai and I want to visit Suzhou and Zhouzhuang too. How can I get there?

Hi, actually, the transportation from Shanghai to Suzhou or from Shanghai to Zhouzhuang is very convenient, both buses and trains. Thus the trains certainly operate between the destinations.
Here is my advice:
From Shanghai to Suzhou, you can take a fast train at Shanghai Railway Sttation. Also you can take a high-speed train or a bullet train at Shanghai Hongqiao Railway Station. The whole journey only takes about 30 min or a little more.
From Shanghai to Zhouzhuang, you can take a direct long-distance bus at Shanghai Long-distance Bus Station to Zhouzhuang. The buses are in operation from 08:00 to 16:50. And it costs about CNY 29 per person.