Wú or "Shanghai dialect" or "Shanghainese" is the main local language for most of Zhejiang province, the municipality of Shanghai and southern Jiangsu province. There are also some speakers in other Chinese provinces. 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, capital of the Kingdom of Wu centuries back, and home to many scholars), not that of Shanghai.
Few travellers are likely to need or want to learn Wu. Throughout China, Mandarin has been the only language of government and education, and the main language of media and entertainment, for over half a century. Nearly all Wu speakers speak at least some Mandarin as well, and younger to middle-aged ones are generally completely fluent. Also, the major cities where Wu is the main local language — Shanghai, Suzhou, Ningbo and Hangzhou — are all quite prosperous and full of immigrants or temporary migrant workers who come from areas where Wu is not spoken. As anywhere else in China, Mandarin is the lingua franca for communication between people from different areas.
That said, if you are spending a significant length of time in the region, then learning some Wu may be useful. It will allow you to join in conversations among the locals, and would almost certainly please and amuse them.