I'm from Tasmania, so I can give you plenty of tips on where to visit! Although given a week you'll barely be able to scratch the surface. I've spent nearly 25 years there, and there are still places I haven' seen.
Definitely spend a saturday in Hobart. Check out the Salamanca market (biggest outdoor market in Australia) and then spend the afternoon and early evening hiking up Mt Wellington. Stunning views.
As for your other six days, well, what can I say? Cradle Mountain is a must see, but then again, so if Freycinet and the Tasman Peninsula. Port Arthur receives a lot of touists, but has a LOT of history and is a really interesting place to visit for an afternoon. Also, you can do a hike to Cape Hauy or Cape Raoul on the Tasman Peninsula. They are both about 4-5 hrs return by foot, however the 250m sea cliffs are just spectacular.
Mt Field National Park is also great to visit for waterfalls and alpine hiking, and you're guaranteed to see hundreds of pandanis

Anyway, send me a PM if you want more details.