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Taxis are omnipresent and cheap, and save a good deal of time and energy over buses. Plus most drivers are friendly, although I have yet to meet one that speaks more than one or two words of english ("hello" and "goodbye", basically).
Buses are great and very cheap. Bus No. 26 is the most popular route between the old and new sections of town, running along the waterfront. It starts near the McDonalds by the train station, and runs along the waterfront past Beach No. 1, past Zhong Shan Park, past the Hai Tian and Shangri La Hotels, past May 4th Square, before arriving at Nanjing Road and Carrefour (half a block from Yun Xiao Road, the best place to eat in town). The bus then runs up Nanjing Road, where there probably isn't much for a tourist to see or care about.
A subway is currently being built, I'll have to give an update after my next trip if nobody else chimes in.
I've walked from the Qingdao Pier to May 4th Square a couple of times, and it's definitely do-able if you've got time and favorable weather. Actually makes a nice day trip, hitting beaches 1 and 2 along the way.