It may be small but there is no way you can see anywhere near all the state in 7 days.
Port Arthur
Mt Field national park (Russell Falls)
and if you continue on from Mt field and go to places like Strathgordon and beyond, you will get to see the tallest dam in the southern hemisphere (Gordon Dam) and Lake Pedder. You can also turn off on the way (but unsealed road) and drive to Scott's Peak inside the South West World Heritage Wilderness.
Cockle Creek and then a 2 1/2 hour walk to South Cape Bay - As far south as you can go in Australia.
Hartz Mountains national park
Richmond - oldest bridge and church in Australia
Peppermint Bay Cruise (from Hobart down the Derwent, past salmon farms, nesting sea eagles)
Any of the Pennicott cruises -The one from Port Arthur back to Eagle Hawk Neck is amazing. So is the one that goes around Bruny Island.
MONA - Museum of Old and New Art - voted best museum in the world.
Ross - second oldest bridge in Australia
East Coast - above what is listed above, go to Bay of Fires. It was listed as the best national park in the world about 3 years ago.
Narawntapu National Park - amazing place, LOTS of wombats
Stanley- The Nut
Queenstown - Lunar landscapes
West Coast Wilderness Railway- steam train ride into the wilderness
Strahan- voted best small town in the world recently ( boat trips down the famous Franklin River into the South West Wilderness
The Bledestowe Estate Lavender farm - call them to see if they are open - largest lavender farm in the world
Oatlands - Australia's only wind mill operated flour mill - it is also the oldest windmill in Australia
Mt Wellington - sunrise is best for photos of the sun coming up over the city - bring your winter wear - even in summer it can get to - (minus) 10 or lower at the pinnacle (eg - its 17 degrees in Hobart right now, but 20 minute ride to the pinnacle, and its -3.7)
Buckland - church has the oldest stain glass window in the southern hemisphere- the window is from about the 13th century, and was shipped from the UK.