Rent a car and drive around the state. Cradle Mountain in the middle, Wineglass Bay on the east coast.
Go to MONA, which Bery posted about.
Between Christmas and New Year there’s a fun festival called Taste of Tasmania at Salamanca Wharf in Hobart. There’s also a market there every Saturday if you want to buy Tasmanian things.
The museum has a stuffed thylacine.
Eat seafood - Bruny Island oysters. Bruny Island is a nice relaxing place to go to if you just want a couple of days of peace and quiet.
There’s an award winning whiskey distillery.
Tasman Peninsula is a nice day trip from Hobart - for the convict ruins but also for the weird geology - coastal formations, dolerite columns, the tessellated pavement, etc. You should also check out Mike Pennicot boat tours.
For weird food, there’s a restaurant that serves seaweed and stuff like that.