Metro Tasmania operate a public bus service in Hobart, Launceston and Burnie. Metro have routes that reach or can drop you near by most if not all major tourist attractions and activities, some walking after that may be required.
An adult fare will set you back $2.80 for a short trip (anything that's around half an hour's trip) and will cost you $4 for anything longer. If you are planning on catching a lot of busses while your here it's worth getting a greencard which will save you time and some money.
Please note you will need to catch the airport shuttle if you plan to get out to Hobart or Launceston airports.