Before saying anything specific, it is worth noting that the culture, expectations and rules varies a lot within Australia. Universities differ; faculties and departments differ within a university. So, it's definitely worth starting conversations with people who might be your mentors at your new university, and reading material on the university website (e.g., grants office, rules about probation and promotion, information about the workload model, etc.).
Funding: Regarding funding, most Australian universities will have internal funding programs designed for different levels of experience. In particular, there will often be programs for early career researchers.
The ARC (non-medical) and the NHMRC (medical) are the two main government funding schemes. They each have schemes designed for early career researchers. They are highly competitive. Your new university should have a grants office, and probably an academic in your school or faculty who can advise you about grant strategies.
You'll want to learn more about ARC Discovery (for more pure science) and ARC Linkage (where you have an industry partner who is providing resources) funding schemes.
You ask about funding "tailored to new staff". If this exists, it will generally be at the university-level. Funding schemes outside the university are generally based on time since you were awarded PhD. So early and mid-career funding schemes will typically look at time since PhD possibly with adjustments for career interruptions and so on. Thus, whether you have changed universities is not really relevant.
That said, universities will often provide additional support to new staff to help them get their research program started. In some cases, you will be given a default allocation of research time which is protected in the first few years regardless of performance, but after a few years, your research allocation may be based on performance.
Probation. You ask:
In the UK most people pass probation (at a much higher rate than passing tenure in the US). Is the same true for the probationary system in the Australia?
Yes. I think that most academics pass probation. The expectations for probation vary between institutions, but it is not like the U.S (although see @daaxix's comments that some Australian universities are moving more to the U.S. model of probation/tenure; I've particularly heard this from those at Group of 8 institutions). The academic hierarchy seems similar in Australia and the UK. In general, the challenges in the Australian system are less about passing probation and more about getting promoted. So for example, each jump from B (lecturer) to C (senior lecturer) to D (associate professor) to E (professor) is where you need to prove yourself. Many So for example, someone who is doing less well with research, teaching, service and so on will struggle to get promoted. More generally, only a few academics get to level E. It is common for academics to plateau at a particular level (e.g., C or D).
In addition, the concept of tenure is different in Australia than in the United States. My understanding is that Australian academics on ongoing contracts and who have passed probation do not have tenure. Rather, they have an ongoing job. Thus, you could presumably only lose your job for performance issues or redundancy. In general, redundancies are quite rare, but they are not unheard of.
Teaching/research/service mix: Universities and departments vary a lot in how they allocate workloads. @daaxix mentions that you may be notionally allocated 50% of your time to teaching. This is perhaps a useful guide, but there is a lot of variability. For example, in my department, the default allocation is 40% teaching, 40% research, and 20% service. The 40% research allocation is protected for your first three or four years. After that, it is based on research performance metrics (e.g., publications, grant money, PhD completions). So, academics who achieve less in the research domain will be asked to do more teaching or service. And if you consistently excel at research, you can get a greater research allocation. More importantly, universities and departments vary substantially in the hours they assign to all these activities. Some universities have very strict and bureaucratic workload models (e.g., you make $49,000 instead of $50,000 in grant funding, then you might find yourself taking extra tutorials, etc.), whereas others are more fluid and flexible.
