I feel there is always something interesting in places where not much expat go... south of temple of Heaven, you have the infamous coton market, muxiyuan, and one of my favorite ever! 
Going south from there, you can find nex to the south 4th ring road a market full of furnitures, including one shop selling amazing tibetan carpets and such. 
There is also a shop dedicated to bamboo items. It's not all good but it's fun. And a kitchenware market, ultra cheap.
As for visits, i've been to the Flower Exhibition Museum, displaying flowers from everywhere. Not much interesting but they do have an amazing selection of orchids.
Just go for it, select an area and explore, i believe it's the best way to find hidden treasures!