Since you ask about Australia, and are living in Australia, you will need to know. A chemist is a scientist with a BSc and PhD in chemistry.
A pharmacist has a BPharm or MPharm (or Cert/Dip. Pharm - but these lower qualifications are being phased out). A pharmacist is very commonly called a chemist in Australia, and a pharmacy is a chemist's [shop].
"I need to go to the chemist's" and "I need to go to the pharmacy" mean the same thing in vernacular AusE.
Non-pharmaceutical chemists generally refer to themselves as "industrial chemists" or even more specific terms, like "polymer chemist".