Beijing has the emperors. Shanghai has the bankers. The port opened to foreign trade in 1843, and for the next hundred years the strip of land along the Huangpu River filled up with banks, hotels, clubs and consulates from a dozen countries. The Bund is what is left of that world, and since the 1990s it has faced a second skyline across the water in Pudong, built by a Shanghai that was no longer taking orders from anyone. The two banks of the river tell the story better than any museum.
It is not the China of temples and dynasties, though there is a Ming dynasty garden in the old city and a different kind of garden culture an hour away in Suzhou. It is the China of dumplings eaten standing up, of plane-tree streets in the old French Concession, of towers that light up at night and a metro that gets you almost anywhere for a few yuan. Five days is a good length: enough for the Bund and the old city, a full day in Pudong, a slow day in the French Concession, a day trip to Suzhou and a last day for People's Square and Jing'an. It is also enough time to get used to a country that runs on Alipay and WeChat, where Google does not open, and where a high-speed train ticket needs your passport.
The plan is built day by day, one area at a time. Day one is the old city and the Bund, day two Pudong and the Shanghai Museum East, day three the French Concession, day four the Suzhou gardens, day five People's Square and Jing'an. Start on a Tuesday: the new museum building in Pudong is closed on Tuesdays (so Pudong goes on Wednesday), the Suzhou Museum is closed on Mondays (so Suzhou goes on Friday, away from the weekend crowds), and day five lands on a Saturday when the People's Square museum is open. For the bigger picture see our China country page. The practical part, visa-free entry, paying, internet and trains, is in the tips at the end and worth reading before you book.