I would suggest Hobart initially. Either stay somewhere around the docks or Battery Point/Salamanca Place or Sandy Bay - which are all within walking distance of the CBD and the most historic parts of Hobart or, for "something special" (depending on your budget) stay at the accommodation chalets at MONA (Museum of Old and New Art), which is situated in a vineyard by the riverside and also has a brewery and resturant on site. (MONA has its own ferry to and from the city).
Around Hobart - Salamanca Market on Saturday morning, the Tasmanian Botanical Gardens, the Tasmanian Museum which has a special focus on Antarctica, the Australian Wooden Boat Festival at the docks (one block away from the CBD) from 8th - 11th February, Louisa's Walk, which is a dramatised history of female convicts in Hobart,
Then use the car for trips further afield....Mt. Wellington, behind Hobart, Mt. Nelson (a suburb which has a cafe with great views at the top), down the Channel area to the Huon Valley, Bruny Island (great sealife watching boat tour), down to the Tasman Peninsular to visit the Port Arthur convict settlement. Also great coastal scenery there and boat tours around the peninsular, Richmond (one of the earliest Tasmanian villages about 20 minutes from Hobart) set in the Coal River Valley with lots of vineyards and cellar door tastings and sales. A little further on Bonorong Park, an animal park with the species which are endemic to Tasmania. If you have more time you can then choose to venture north along the east coast or head west towards the wilderness and Strahan and Queenstown.
I think you would really need another week for the north of the state.
I would recommend you have a look at these sites which have plenty of suggestions including, on the second one, some suggested itineraries.
http://www.tripadvisor.com/Tourism-g...Vacations.html
http://www.discovertasmania.com.au/
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