Many governments have extensive websites (though keep in mind they won't always be in English). However, these websites can often be complicated or difficult to search.
One way to search government websites easily is to use the Google site: operator.
A search for election information from the Australian government would look like this:
Australian elections site:.gov.au
The first part is your keyword (Australian elections). The second part tells Google to only return websites with the domain .gov.au, the domain used for Australian government websites. Make sure there is no space between site: and the domain.
See below for a list of other international government domains.