Best bet would be to head up the Mt Anne Track, maybe as far as the High Camp Hut - good track, easy to follow and amazing views.
After High camp hut you will need skills on rock to go to the summit of Mt Eliza. I did this walk several times when I first arrived in Tas, progressing further the more confident I got.
Pretty steep, but you can turn around any time you want.
Here's a pic to whet the appetite!
Taken half an hour from carpark. It gets better the higher you go. but I won't spoil it with more pics
Port Davey Track will give you good views of the Western Arthur range and is a lot flatter - once again it is a go as far as you feel comfortable, then turn around and head back.
P.S. Welcome to Tassie!!!