If you're interested in sharing transport or sightseeing with other people, look at the page Sightseeing With Other LCAers
Half-day trips, close to Hobart
There are lots of great walks close to Hobart. Mt Wellington would be a good start, walks range from 2 hours to 5 hours, with varying levels of climb. http://www.parks.tas.gov.au/index.aspx?base=1484
For Day trips, short drive:
Douglas-Apsley National Park
2 hours drive out of Hobart, no idea about how long any walks there are (Close to Wineglass Bay, closeish to St Helens).
2 hours out of Hobart, plus maybe an hour's walk to get to the Bay -- I've done the walk, it's quite worthwhile to go right around Freycinet Peninsula, it's more like a day walk if you want to do that. (Close to Douglas-Apsley, closeish to Port Arthur)
1 hour out of Hobart, definitely worth going to (and quite easy to get to) (Closeish to Wineglass Bay)
Is 1 hour out of Hobart, and has walks ranging from 20 minutes to 6 hoursish, plenty of nice waterfalls to see, if you like waterfalls. http://www.parks.tas.gov.au/index.aspx?base=3589
Huon Valley, Southern Forests (Hartz Mountain & Tahune Airwalk), and Hastings Caves
You can get to Hastings in ~90 minutes, and are quite worth visiting (quite pretty caves), the Hartz Mountain National Park has some reasonably-lengthed walks, which you may enjoy. There are also plenty of fruit farms on the way (near Huonville) which will sell you apples or cherries for a decent price -- cherries in particular are in season at the moment, and so if you buy those, they'd be practically straight off the tree.
Overnight-2 night Stay:
4 Hours out of Hobart, plenty of things to see there -- the Wilderness Railway and any Cruise up the Gordon River are particularly worthwhile.
Bay of Fires
3-4 hours out of Hobart, apparently very good -- Lonely Planet reckons so, anyway! (Closeish to St Helens)
3ish hours out of Hobart. (Closeish to Bay of Fires)
Seven Sheds Micro-Brewery, Meadery and Hop Garden http://www.sevensheds.com/
(Railton, half-hour south of Devonport)