No, you cannot travel to New Zealand with an Australian student visa. You must apply for a separate New Zealand visa, such as a visitor or student visa, depending on your purpose of visit.
With an Australian student visa (subclass 500), you can travel in and out of Australia as often as you require, as long as the visa is valid and you adhere to the conditions.
After obtaining an Australia PR, you can travel to New Zealand without requiring a visa. As a permanent resident of Australia, you can visit, live, and work in New Zealand.

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What can I do with an Australian student visa?
An Australian student visa lets you do the following activities. You can:
- Live and pursue your studies in Australia.
- Travel freely in and out of Australia with a valid student visa.
- Work for a maximum of 48 hours per fortnight
- Apply for work in Australia after completing your studies.
- Sponsor your family members to come and live with you in Australia.
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How to travel to New Zealand with an Australian student visa?
Australia and New Zealand have distinct immigration laws. Having an Australian student visa does not grant you access to enter or live in New Zealand. If you want to travel to New Zealand, you must apply for a New Zealand visa. It can either be a visitor visa for tourist activities or a student visa for study purposes.
If you require a visa to enter New Zealand, you must apply for an NZeTA (New Zealand Electronic Travel Authority) before traveling to the country.

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Steps to apply for Australia PR as an international student
As an international student in Australia, you can apply for PR after you graduate from your study program. You are also required to demonstrate your skills and qualifications through a Skills Assessment, fulfill the requirements for a skilled visa, and submit an Expression of Interest (EOI) through SkillSelect.
The detailed procedure for applying for Australia PR as an international student is given below.
Step 1: Complete your study program in Australia.
Step 2: Apply for a post-study work visa to extend your stay in Australia and work.
Step 3: Have your skills assessed in Australia for your nominated occupation.
Step 4: Submit an Expression of Interest (EOI) through SkillSelect.
Step 5: Obtain a PR and travel to New Zealand from Australia without a visa.
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