New South Wales

TAFE
TAFE colleges offer a wide range of courses including basic IT courses
http://www.tafe.nsw.edu.au
or call the TAFE Information Centre on 13 16 01.

Adult and Community Education
Board of Adult and Community Education (BACE) provides a list of provider contact details.
http://www.bace.nsw.gov.au/index/contents.htm

Community Colleges website list short courses throughout NSW.
http://www.acensw.com.au

Other providers
Many Business Enterprise Centres also run short introductory courses on IT for small business.  Call +61 2 9338 6660 for the nearest Business Enterprise Centre location, or see the Business Advisory Service Centres page on this website.

If you can't find what you're looking for here, you might like to explore some of the Australia-wide training directories which may cover courses offered in your location.