My favourite websites for gathering information:
*Note that birth/death/marriage records and similar dated records are very likely to contain name variations, so do not settle on looking for just the current spelling of the name. If a widow remarries, her surname is listed as that of her previous husband, rather than her maiden name.
familysearch.org The best resource for accessing original Tasmanian records on my opinion (far better than Ancestry and the other paid sites that mostly only offer basic transcriptions). Don’t rely on the name search, you may need to go through the documents individually to find what you are looking for.
LINC Tasmanian Names Index While not as good as familysearch.org, it has a much better name search function and offers some records which are not available on familysearch. *Note that more recent Tasmanian death and marriage records can be purchased, however, there is no publicly available index, making it difficult to know what you are searching for.
Victoria Births Deaths and Marriages Indexes are not official records, but they are a good way of double checking information such as a parent’s name *Note that transcriptions can be wrong. You have to pay a hefty sum to actually access records.
NSW Births Deaths and Marriages Again, not very good in the fact that you have to pay for records, however, the searching features for this index are much better than the Victorian counterpart. *Note that they butcher the official records somewhat. I payed for a record that they had somehow altered to say that a marriage occurred in a Church, when the original records states that they were married in a house.
Ryerson Index (Index of Death Notices in Newspapers Across Australia) *Currently missing ‘The Advocate” which covers north-west Tasmania. Members of the LINC library can access most issues of the Sydney Morning Herald for free.
Trove Digitised Newspapers: Australian Newspapers up to 1955. *Note you can manually search through the papers to find family notices for a given issue rather than relying solely on keyword searches. This is more reliable than using keywords because many of them are not picked up.
Canberra Veterans Name Search A mixture of free and paid records. WWI records are free and I believe that WWII records become free to everybody once somebody has paid the fee for that record.