Heading to Oz in May/June is getting towards low season and you couyld use the Boots search for fares to compare with what else you can find available with the likes of
www.mobissimo.com kayak or travelocity etc.
You'll have to fly into one of the mainland capitals, Sydney or Melbourne probably best as Tasmania does not have international airports, but many daily connections from Syd./Mel.to Hobart and lesser to Launceston or Devonport.
To check that you do not get ripped off for an add on,
www.webjet.com.au will give you Oz internal flights, and often good specials on, and Syd./Mel. to Tasmania can be obtained for under $100.
You could also take the ferry from Melb. to Devonport on Tasmania north coast for about the same $ as a flight, ferry takes about 12 hours as an overnight trip.