The airport is about 16km east of the city centre, and to travel between the two you can hop on the TRC shuttle service or take a taxi.
In the city most buses leave from Elizabeth Street offering various tickets. For unlimited all day travel a Rovercard is a good idea, which can be used at anytime during the weekend and from 9.00am - 4.30pm and after 6.00pm during weekdays.
There is also a ferry service that offers transport to a number of sights around Hobart working on a hop on hop off basis. The Wrest Point Wanderer travels between 10.30am - 3.00pm. Ph: 6223 1914. You can also take boats further afield to Flinders Island and Port Welshpool in Victoria.
You can also get further travel information and contact numbers for the whole of Tasmania, from the Tasmania Travel Section.