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So pretty much YuYuan Gardens, TV Tower.. stroll around the old town? If there are any old farmer’s markets, or perhaps flea markets, that would be interesting as well.. but really, I’m just looking for some advice as to what to see in Shanghai, generally.
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Come for the food then. 
The old town is in the Yuyuan neighborhood. The French concession area is an old town but different part of shanghai and different thing altogether with old villa’s.
If you like to eat, and you like to eat in restaurants in old historic buildings, then Shanghai is paradise. There’s tons of great restaurants that are pretty reasonably priced, and even most of the expensive ones are not bad by Singapore standards. An example, Shanghai has not one but two continental restaurants in formerly russian orthodox churches. Japanese style european food in a former villa. Villa restaurants are some commonplace that I can think of at least a dozen without trying.
Skip the TV tower. if you must have such an overview of Shanghai and the weather is clear for same, then go to the bar on the 87th floor of the Jin Mao Tower (Grand Hyatt hotel). IIRC it opens at 18:00, and you can either make reservations or get there when it opens for a great view and a drink. They have some good restaurants on the 56th floor or so that also offer a great view.
One of the river towns would make a good day trip. Probably more like a bazaar, flea market, chinese tourist crap mixed in with a historic setting. Better than any market in Shanghai, that’s for sure. Some of them have pretty cool peasant art.
Most people like Hangzhou, especially if it’s kinda arranged for you and you get a good restaurant out of it. I hear there is a new fast train between Shanghai and Hangzhou that makes it pretty convenient. If you want a buddhist temple, definately Hangzhou (IIRC lingyun temple) is THE place instead of the really lame ones in Shanghai. Food’s good too.
You’re in the right season for Shanghai hairy crab. You don’t have to go to Bacheng on the Yancheng Lake, but that can be a good experience. Again, need to have it arranged to go to the right “floating” restaurant but it’s pretty good. You can get great hairy crab in Shanghai too, but something about having them pulled out of Yancheng Lake a few minutes before being steamed up is just heavenly. Here’s a hairy crab and after steaming and opened. It’s not for everyone, but it’s damn good.