The best things here are free and indoors — the museums — or free and outdoors — the streets — and what you actually pay for is height, so save the tower for a clear day.
Shanghai Museum East
The museum's Pudong building at 1952 Century Avenue, open Wednesday to Monday 10:00–18:00 with last entry at 17:00 and closed Tuesdays outside national holidays. Individual admission is free and needs no reservation — bring your passport and walk in. Metro line 2 to Shanghai Science and Technology Museum, exit 8, or line 6 to Pudian Road, exit 1.
Yu Garden
Classical Jiangnan garden begun in 1559, in the old city at 168 Fuyou Road, open 09:00–17:00 in peak season and 09:00–16:30 off season. The garden proper is ticketed and calm; the bazaar of shops and dumpling stalls wrapped around it is free and heaving, so do the garden first and eat afterwards.
Top of Shanghai Observatory, Shanghai Tower
The observation floor on level 118 of the Shanghai Tower in Lujiazui, 546 metres up, reached by high-speed lifts. It is worth booking a slot for a clear afternoon and staying through sunset; on a hazy day you are paying to look at cloud. Metro line 2 to Lujiazui.
Power Station of Art
Mainland China's first state-run contemporary art museum, converted from the Nanshi power plant and opened on 1 October 2012 at 678 Miaojiang Road on the Huangpu. Its 165-metre chimney is itself an exhibition space with a rooftop terrace over the river, and it is the home of the Shanghai Biennale. Special exhibitions are ticketed separately.
Wukang Road
A curving street of 1920s and 1930s villas and lane houses in Xuhui that you can walk end to end in under an hour. The writer Ba Jin's former residence is at No. 113, Ferguson Lane at No. 376 has cafes and galleries in a courtyard, the 1932 garden villa at No. 390 is worth a look, and the tourist information centre at No. 393 gives out a walking map.