If you're interested in art you can easily while away half a day or more at Moganshan Lu near Suzhou Creek, which is an old industrial complex that's been converted into gallery space. MOCA and the Duolun Museum of Modern Art are also both well worth checking out.

As far as museums go, it depends what kind of thing you're after. There's the Shanghai Museum which is impressive but if you're looking for something a little more offbeat then how about the Propaganda Poster Museum?

For a break from the city, there are several "watertowns" such as Zhujiajiao, scattered around the outskirts of Shanghai that make a nice day trip and give a taste of the Old China. These Ming and Qing-era villages are built around networks of canals with traditional whitewashed houses, cobbled streets and shaded courtyards are quite charming. The weekends can get quite busy with tourists but still, it's a welcome break from the skyscrapers of Shanghai. Though not far from Shanghai, the easiest way to visit is probably to go with a tour. Ctrip.com offer a few day and half-day trips that might be interesting:

http://activities.english.ctrip.com/Activities/PkgDetail.aspx?pkg=19127&city=2&from=2010-7-15&to=2010-7-22&aduNum=2&chiNum=0 for an idea of itinerary and price.

You can find plenty of info on all of the above at http://www.chinatravel.net/home/

If you like art, you might consider just taking a walk along Shaanxi Road, the southern end has got lots of galleries and some antique shops there too.

Otherwise I would recommend a dive in the shark tank at Changfeng Ocean World. No, I'm not kidding, you can swim with sharks!

http://www.chinatravel.net/china-attractions/changfeng-ocean-world--aquarium-/introduction-1191.html