Hi Phil,
We did Tassie on our trip back from Cairns to Perth early last year (2016). We also planned 3 weeks and booked the Spirit of Tassie accordingly but even though I'm a Taswegian and SWMBO has also lived there we ended up needing 6 weeks !!! It was not easy to change the ferry booking at that time of year but we were lucky eventually. As far as must do's go, here is my list:

1. MONA (Museum of Old & New Art) - this is hugely popular and a definite must do and its close to Hobart city.
2. The Wall in the Wilderness (sometimes just called the Wall) - it's a massive Huon Pine carving at Derwent Bridge (on the way from Queenstown to Hobart).
3. Salamanca Markets (Saturday morning) - they're right on the city waterfront and sell everything (make sure you buy a scallop pie).
4. Port Arthur (of course ) - but try the Tasman Island Cruise (3 hours from Eagle Hawk Neck to Port Arthur - they bus you back to your car) - this can be the best day of your trip, especially if the conditions are good, but its not cheap.
6. Tamar Valley wineries - try Tamar Ridge, Pipers Brook, but there are plenty and this region produces some of the best Pinot Noir in Australia.
7. Strahan - on the west coast. It's beautiful and gives you access to the Gordon River World Heritage area.
8. The Tahune Airwalk (about 30km from Geeveston) - the view over the Huon River is amazing
9. On the East Coast - Freycinet National Park & Wineglass Bay. Beautiful.
10. Mount Wellington - great views back over Hobart and the Derwent River and it's so close to town.
11. If you have time to get into the middle - do Cradle Mountain & Dove Lake - they're an icon of the state.

There is so much to do in Tassie and there are heaps of great free camping spots, so make sure you have Wiki-Camps or the Camps Australia book.

Enjoy
Creedo