We've got a great free brochure in the shop called Sharing the Road in Tasmania - a guide for drivers, cyclists and pedestrians. There's lots of great information for all road users in this brochure so drop in a pick one up next time you are passing the shop.
Below is an except from the brochure covering some important road rules (and fines) correct as of August 2006:
Going through a red light.
Fine: Motor vehicle $110 - Bicycle $80
Insufficient lights at night (white front, red rear)
Fine: Motor vehicle $110 - Bicycle $80
Using a handheld mobile phone while driving / riding
Fine: Motor Vehicle $110 - Bicycle $80
Fail to give way to a pedestrian on a pedestrian crossing
Fine: Motor Vehicle $250 - Bicycle $110
Causing a hazard by opening a car door into the path of traffic (including into the path of a person riding a bike)
Fine: $110
Overtaking too close to another vehicle
Fine: $110
Driving without reasonable consideration for other road users
Fine: $80
Not wearing a bicycle helmet
Fine: $80
Cyclist of driver leading an animal while riding / driving
Fine: $80
Riding motorcycle / bicycle 3-abreast or more
Fine: $80
Cyclist failing to give way to a pedestrian on a shared path
Fine: $50