Tassie at Christmas.

On the west coast you can drive the beach to Sandy Cape (not a trip on your own)
or take the inland track to the Pieman River X-ing. A good drive to the top of Mt Balfor from there you can see the inland track, you can get to Balfor from the coast via the old rail line. Not bad camp sites down that way as well.

See the rangers at Mt Field and get the key to the old copper mines, a track that requires good clearance as the wheel ruts can be a bit deep through the mud.

A drive to the Pieman Heads from the south is another good track which follows the beaches (just watch the waves and the bog holes)

The button grass swamps down south can give you a bit of fun as there are some hugh bogs on the way to the costal camp sites

Andrew,
Did that trip last Xmas (also from Brissy) Had a good time but remember to take the woolies.
Don't forget the Dubbo zoo on the way there.
The Holy Cow cafe and cheese factory and the Pub In The Paddock, both at Pyn Ghana (spelling?) on the highway between St Helens and Scotsdale.
The Elephants Pass cafe makes the best crepes you've ever tasted. (no EFTPOS so take your cash).
Mt William National park in the north east has some good campsites near the beach.
Port Arthur area.
Down south Hastings caves are amazing.
Up north, take a drive up Mt Ben Lomond. JACOBS ladder is unreal!! (not for trailers)
Russell falls national park, good camp ground and good forest walks, only short distance from the Styx valley (giant old growth forest)
West Coast road from the Pieman up to Arthur river is a good dirt road. Can get very wet and slippery at times.
Gordon river cruise.
Cradle Mountain has some great walking tracks but only for well prepared experienced hikers.

Enjoy your trip

Another one is the tree top walk at Huon & trian from Strahan to Queenstown. We spent a week down there last Nov & saw bugger all ,not enough time, but will go back .