Some of these are even accessible by metro...
Chongming Island: - Really chill island with wetland parks, bike riding, and solid home-cooked food.
Binjiang Forest Park: Light camping, BBQ zones, a pretty mental experience.
Fengjing Town: The water town you haven't heard of. The place where all the xiao long baos come from.
Daguanyuan: Red mansions, grand views, water, pagodas.
Zhouqiao: A temple museum, pagoda, gardens, and quaint streets. Accessible by metro.
Links Hot Springs Resort: Just a hotel with some hot springs and golf courses in Jiangsu. They might have some deals on rooms as it's not that cold yet.
Qingdao: Ok this is actually a six-hour train ride from Shanghai, and you'd probably wanna wait until a few days into the holiday to take a train anywhere, but Qingdao is great. Seafood, old European architecture, clean air, beaches, and chill folks. And beer.
Just go somewhere. Really -- your sanity depends on it.