This little yellow bus/train? carries tourists around the town. I didn't take it, so I don't know how long the trip takes or how much it costs, but it seemed popular with older tourists, especially if it's a wet day.
Quedlinburg is about an hour's train ride south west of Magdeburg where you have to change trains when coming from Berlin (a three hour journey). If you want to get here from the rest of Harz it is only Wernigerode which is easy to reach on the same line. For Goslar, you have to go by local buses since the railway obviously never crossed GDR borders so there is none here.
Once in Quedlinburg (and Wernigerode nearby), you can go on steam engine rides in the Harz valleys and up mountains. Use the second weblink below to see where you can go or visit the HSB exhibition near the Markt in Quedlinburg (children will love it since there is a model train).