You know, once I thought these places in Tassie were pretty secret. Then I went to Azores in the middle of the Atlantic Ocean and gave a lift to a 16 year old hitchhiker who was going to the other side of the island to go body boarding and the first thing he said after I said I was from Tasmania was "ahh shipsterns" :O
Two very well known places are Marrawah on the West Coast and Shipsterns in the South.
That surf forecast site has plenty of places to surf all over Tas and Aus. about 8 years ago
Not much of a surfer myself, but I have quite a few friends back in Tasmania who are well and truly into it. Some of the best spots in Tasmania can be found along the West coast, which can be a pain if you are a tourist as these areas can be quite hard to reach without renting a car. Still, if you can get some like-minded people together, it can also be a recipe for an awesome road trip!
The above is Marrawah, on the northern end of the West coast. Some of the best waves in the state can be found here, so I am told.
Closer to civilization is Clifton Beach, about 30 minutes from Hobart. You may even be able to get a Metro bus out this far, check www.metrotas.com.au for more details.
EDIT: Thanks for pointing out Shipsterns Niko, completely forgot! about 8 years ago