The best way to get around Launceston is by car. Taxis are available, may add up to be expensive, and try to get a transfer when you are moving from the airport to your hotel. Car rentals are plenty, from Budget, Europcar, and the rates are great. There is also a Metro bus service which goes all through the town.
Bicycle is a reasonable way to get around town and out the Gorge. Traffic is fairly quiet in town, and there is a good network of bike paths.