Sorry, I should have been more specific on web sites.
Ancestry is http://www.ancestry.com.au a subscription site and you may have free access at your local library and/or Family History Centre (local Mormon Church).
FindMyPast is http://www.findmypast.com.au, another subscription site and once again your local library or Family History Centre may have free access.
TROVE - the National Lirary of Ausralia's wonderful website of old newspapers - searchable with images - http://trove.nla.gov.au/newspaper.
The LDS site has a lot of helpful tutorials on family history at http://www.familysearch.org. They have some Australian records in their collections but many more US and Europe.
For a bunch of useful links try Cyndi's List at http://www.cyndislist.com/australia.
A word of caution: there are a lot of family trees out there with dubious if not outright wrong data. Take these with a grain of salt and use them as hints of where to look for your folks.
Before going any further, has anyone else worked with Thomas Burgess? No other trees out there? Google him with variations of his name. Who knows what you'll find.