Zhong Shan Road has a lot of small shops and boutiques, plus a lot of tourist-oriented shops.  Spend a day walking the full length of the road and you'll get a variety of experiences.

Jimo Lu is the primary bargain market in Qingdao, the best place to find all the cheapest goods, knockoffs, bootlegs, etc.  It's located near the far end of Zhong Shan Road, but any cab driver will recognize "Jimo Lu" and get you there.

Cultural Market is a little artist colony in Qingdao, where you get great deals on art, antiques, etc.  This is the same stuff they sell in the big Beijing "Friendship Stores" at about 1/10 the price, and open for heavy bargaining.  The market is hard to find, however, and most taxi drivers don't know it.  It's located near the Tai Dong district, actually just a few blocks behind the old Tsingtao Brewery.  This street is currently undergoing a major renovation, like everything else in China probably targeted towards 2008.