>>2636618The obvious one is The Overland Track. Scenery is spectacular and there are heaps of peaks to do as side trips.
Problem is it has become super popular and they now have a ticket system so it can be difficult to get access. You can do parts of it without a ticket though.
Cradle Mountain/Barn Bluff can be climbed as day trips or an overnighter, and look up the Arm River Track to get into the centre portion without an overland track ticket.
The Overland track has a really well made track by Tassie standards. Don't expect much difficulty navigating etc, will likely still be mud and plenty of tree roots though.
Frenchmans Cap is the next step up in difficulty. If you want to make this a bit harder add either Clytemnestra or Mt Philps+Agamemnon to the trip. Both require off track navigation skills, Frenchmans has a decent track all the way up.
The Mt Anne circuit is also really good, harder than the ones mentioned above, lots of boulder scrambling.
If you're up for more of a challenge go for the Western Arthurs. Spectacular but a bit more hardcore although it is still tracked.
There are endless other ones but as a non local you might struggle to find access info as most dont have 'official' tracks or tracks at all. The ones above will give you a great time and planning should be easier with plenty of info available online.