Located only a couple blocks away from Nanjing Lu, People's Square is a perfect place to take a stroll in the early morning or in the evening on a hot summer's day.
Buy a giant octopus kite or head over to Shanghai's Art Museum if the heat is simply too much for you. The Museum is beautifully built and has several floors and rooms that are dedicated to different exhibits. A personal favorite is the exhibit featuring carved jade; ranging from incredibly small simplistic pieces to extravagantly large and detailed jade jewelry.
The museum also has small shops (although you can find similar items for a much lower price elsewhere), rooftop restaurants, and cafes that serve tea and desserts.
Tip: The museum is not free but well worth the visit if you're a history buff or simply enjoy an air-conditioned building full of artifacts.