We have a few option to stay in Australia. The easiest way is to come here on tourist visa. It is easy and convenient, but doesn't allow us to work. Most European nations can get a tourist visa within 24 hours and it's free of charge for up to three months. Start with the online application.
If you really want to feel the Australian atmosphere you need to live and work in one place for longer. That is possible in other visa's type.
Working holiday visa is the most popular, but not all nations are allowed to use this option. Usually is valid for one year but you can extend it by working on farm (fruit picking or similar) for at least three months. Only young travellers under 30 years old can apply for this visa. Your work is limited to stay with one employer for no longer then 6 months.
There is a bit more research to do with student visa, more detail here.
Sponsor visa. You can try to find a company to employ you. Disadvantage of that one is that you must stay with your employer for longer time, doesn't matter what your relation is. If you are fired it might be that you are going home.
Permanent residency is also available, but it's not easy. I am not able to describe it, you just have to spend hours searching for accurate information on the official Immigration site, or go to the lawyer.
Generally the idea is to get the required number of points - your education, job skills (only selected proffesions give the right amount of points), work experience, the language test (IELTS), age, family in Australia and other possibilities. The lawyer knows the stuff and can help you find the best way. When we find the way, then start collecting the documents, translate them into English, do medical and language test - when all done, just apply and wait for a decision. Sometimes even a few years.