Although there are branches around the world, the Din Tai Fung locations in Shanghai (there are multiple) have amazing black truffle versions of the classic soup dumpling.
If you want Western food, I recently dined at 8½ Otto e Mezzo BOMBANA which had fantastic Italian.
If you're planning far enough in advance, I'd try to snag at table at Ultraviolet by Paul Pairet - it's an experiential restaurant that is the only one of its kind in the world and also on the World's Top 50 Restaurants list. If you can't make a reservation there, you can dine late night (they only open after 10pm) at Mr & Mrs Bund-Modern Eatery by Paul Pairet
The local Shanghainese are really into the new Shanghai location of Da Dong Duck, another branch of the famous upscale Peking duck restaurant in Beijing. Remember to reserve in advance.
If you find yourself having a late night and want a solid bite, Bellagio (there's multiple locations) is open until 4:30am and has awesome Taiwanese staples.
near Zhongshan Park in Shanghai is an excellent Yunan restaurant.