Hi and welcome to Just Answer!If you want your Australian corporation to perform business activities in California - yes - it should be registered - and will be liable for federal and California state taxes.
You may find registration requirements here - http://www.sos.ca.gov/business/be/faqs.htm#form-question8
File a Statement and Designation by Foreign Corporation (Form S&DC–S/N [for foreign general stock or nonprofit corporations], Form S&DC-PC [for foreign professional law or accountancy corporations] or Form S&DC–INS [for foreign insurer corporations) and attach to the completed form a valid certificate of good standing by an authorized public official of the foreign jurisdiction under which the foreign corporation is incorporated.You will also need to obtain a tax ID - EIN from the IRS - see details here - http://www.irs.gov/Businesses/Small-Businesses-&-Self-Employed/Employer-ID-Numbers-(EINs)-Every year - you will need to file federal and state income tax returns for your corporation and will pay California franchise fees.
If your Australian corporation doesn't actually perform any business activities - you might want to register a new corporation or LLC in the US - and woudl not need to register a foreign entity. In some situations that mighty be more simple solution.Let me know if you need any help or clarification.
If you still decide to use a foreign corporation - it will need to file a federal tax return as a foreign corporation - using form 1120F - http://www.irs.gov/file_source/pub/irs-pdf/f1120f.pdf A foreign corporation files this form to report their income, gains, losses, deductions, credits, and to figure their U.S. income tax liability.