For a far more extensive list of events than I can maintain, please check out the events page of the Australian Mountain Bike Association.
Read about the Greater Hobart Mountain Bike Masterplan. If you click "bike" in the left navigation, you'll also find pdf maps of the area.
the Glenorchy Mountain Bike Park (GMBP), located behind Tolosa Park in Glenorchy, has developed opportunities for many levels and styles of mountain bike riding, and provides a quality facility for local and interstate riders. The park has been successfully hosting rounds in the National series.