Like so many things about the EU, it's complicated.
The non-EU partner will usually be required to apply for a residence permit for the country where they are residing with their EU partner, and this can take several months, for example in the case of Italy.
The situation that will apply most often to Australians is where they are married to a UK citizen.
They can then apply for a UK Spouse Visa, which gives them the right to remain and work in the UK.
However, when travelling through Europe, since they do not have a spouse resident permit granted by a Schengen Area country, the Australian partner will be restricted to the rule that applies to all Australians in the Schengen Zone, namely the right to enter and travel freely for 90 days in any 180-day period.
If they both intend to live outside the UK, in France for example, the non-EU partner must apply for a French residence permit.
Once granted, they can travel freely without a time limit in the Schengen Area when accompanying their EU partner.