Exactly what is Audio Engineering?

The term “audio engineering” is very broad, and can encompass a wide range of activities in the audio sphere.

The loose terms “recording engineer”, “live sound engineer”, and the more generic “sound engineer” are used to denote practitioners of the discipline “production and operations”.
These are the people whose skills and creativity are used in the creation of audio content across a wide range of platforms. Here the “engineer” title is historic and does not imply any level of education or qualifications.

This is vastly different from the Engineers working in the more “technical” side – who design, construct, and maintain the equipment and systems used by the operational practitioners. They are typically degreed Engineers in the Electronic Engineering field, or Engineering Associates (para-professionals) who hold Diploma or certificate level qualifications.
Unfortunately both the operations (creative) and technical staff are often, confusingly, bundled into the same “Audio Engineer” category.

In the operations area, there are also other designations, like the film-based occupations of sound designer, film location sound recordist and sound mixer, as well as ADR and dialogue & effects editors.

On the “technical” side, there are also practitioners in the communications, telecommunications, and computer systems engineering disciplines.

The skill sets required to operate in each of these distinct areas is quite different. The operations and production skill-set is strongly anchored in the creative arts. The skill-set in the Engineering disciplines is strongly anchored in science, technology, engineering and maths (STEM). The ways of thinking and frames of reference are (generally) quite different.

If somebody wishes to undergo education in the field, their path will depend upon whether their interest is in the Production (“Creative”) side or the Professional Engineer/Technician (Design/Construct/Maintenance) side.

We cannot offer career advice, but would suggest that if the Professional Engineer aspect is the interest, then tertiary educational institutions offering Electronic Engineering courses be consulted.

If it is the creatively-oriented Production aspect that is aspired to, then there are Colleges that run courses in this field.
Again, we cannot recommend any particular institution but there are several in Melbourne (and other cities) that could be evaluated for suitability to their needs.

A simple web search will bring up a list of candidate colleges.

As a starting point, this Google search uncovers some examples based on Melbourne locations: Eg:

There are other less formal paths to a career, particularly in the Production side – such as mentoring, internships, networking, on-the-job training and the like.
Those are entirely up to the initiative of the individual and their advisers.