Most residents of Australia and New Zealand, with the exceptions of the Aboriginal and Maori people,  arrived as immigrants.  The main means of transportation for those immigrants was shipping.  If you know what year, which ship, port of embarkation, or even just the Port of Entry, then your task of tracing your ancestor will be much simplified. Checking  the Ships Lists should be your first step.
For assistance in identifying which ship your ancestor travelled to N.Z. on, see http://freepages.genealogy.rootsweb.com/~nzbound/

Read a guide on Australian Genealogy - it will give lots of hints.  Also see whats at the National Australia Archives.

Remember to utilise the Family History Centres. If you're not sure where your nearest one is, check HERE.  If you need a tutorial on using the LDS on-line Library Catalogue, check the wonderful tutorial prepared by Le Geefted One.

Don't forget the value of Search Engines. If you find which ship your ancestor arrived on, do a search on Google to get more information about that ship - facts such as size, when built, time of journey etc. fill in the bare bones of your family history. You might be lucky enough to find a photo or plans.

Check the war service records of the one million people who served in WW2.

Lookups on Australian civil and parish registers, and New Zealand civil registers, can be requested from HERE.

For New Zealand genealogy, check out the KiwiGenWeb.     Another excellent site is PearlsPad which has lots of information and names - especially associated with Auckland, but some nationwide also.

The National Archives of Ireland has provided a database of people who were transported to Australia for a variety of crimes and infractions. There are also some 'free settlers' included. Those 'free settlers' were members of the families of those transported.

http://www.censusdiggins.com/freebies.html    Has free stuff for genealogists.