Braunlage is the main base for mountain biking in the Harz. There is a cable car uplift and they have a freeride course back down. Most use it as an uplift to head in the direction of Schierke (also a good mtb friendly campsite here if you are camping). Lastly there is Brocken, the highest mountain in the region. No particularly good trails up and down it but you get some fantastic views or you could use the narrow gauge railway to take you to the top. It gets really windy up there so take a jacket even if it’s nice in the valleys. For touristy things then towns like Wernigerode are worth a visit with their Tudor style houses. HTH.
