Hangzhou (Zhejiang Province), former capital with a famous lake, one of China's top destinations for domestic tourists. Marco Polo wrote of Hangzhou the city is beyond dispute the finest and the noblest in the world.
Nanjing (Jiangsu Province), capital under several dynasties, most recently 1912-1949
Ningbo (Zhejiang Province), a major trading port in the tea clipper era, still an important port and industrial city
Shanghai, China's largest city and greatest economic center; in some ways, the cultural capital as well
Suzhou (Jiangsu Province), old city of canals and gardens, now a major center for high tech. The Chinese have a saying that heaven has paradise but Earth has Hangzhou and Suzhou.
As anywhere in China, Mandarin is the lingua franca; nearly everyone can speak it. As elsewhere in China, English is not widespread but some people speak it quite well.
The region does have its own language, called Wu or Shanghainese. This is a populous region and the number of Wu native speakers is large; at 78 million it is rather more than French or Italian. There are local variants of Wu; the standard is that of Suzhou (an older city and home to many scholars), not that of Shanghai.
Perhaps the commonest way to reach the area is to fly to Shanghai. There are two airports. Pudong is a major international airport with connections all over the world; Hong Qiao handles mainly domestic flights.