When we moved, the following are the first items we did (we were on a 457 visa). Hope this can somehow help those who will be doing their big move soon. =)
Open a bank account - We opened it online while in PH. When we arrived in OZ, we activated the account by bringing the passport when we visited the bank. Any additional requirements (like TFN) can be submitted later on. We opened accounts with ComBank and NAB. When we have settled, we realized that ING has the highest yield so we opened an account online (no need to go to the bank as all their transactions are online). If you are meeting their criteria you can earn 2.8% interest per annum as compared to 1.8% with other banks.
Get a TFN (Tax File Number) - This is needed when you open a bank account but the information can be sent once you have it (so not to worry too much so long as you will send it later on). Your employer would also need this information as part of payroll processing and superannuation contributions.
Find a place/house to rent - We were given 1 month accommodation when we arrived and 3 weeks into AU, we got help form an agent for 1-day home search. We viewed 5 properties and when we have decided on our top 2, we submitted an online application and received confirmation. We paid the bond to close the deal and moved to the place a week after. When paying the bond, we had to request for a check from the bank. The process may vary depending on the agency/landlord's requirements.
Electricity/Gas - We just researched online on the providers and called one that we thought would be good and got connected on the day we moved into the place we were renting. Electricity/gas bill is every quarter. In our unit, there's no gas and the unit has electric stove. Our neighbors recommended that we buy a campmaster butane grill stove to save on electricity. Buying the butane is way cheaper than using the electric stove for cooking. If your unit is also using an electric stove, you may want to consider using the campmaster.
Groceries - If you live in a nearby
Furniture and Appliances - If you are renting a fully furnished house, you won't need this. However, most properties being rented are bare so you need to buy a few items to ensure you have the basics. We bought our furniture at Fantastic Furniture (as recommended by the agency who helped us for our home search). We also bought some appliances like washing machine and electric fan (for the summer heat). We were fortunate to get 2nd hand appliances from relatives and friends like TV, fridge, dryer, rice cooker,etc (saved us heaps of money).
Childcare/School Enrollment - When we arrived, our child is 3.5 yrs old. We had to look for a childcare so that we can both (husband and I) work. It took us about 4-5 months to get a slot because there were lots of kids availing of childcare in our area. This might be shorter/longer depending on your location. There are instances when slots are not available so you need to apply for their waiting list (we had to pay $50 in one of the centers and another one is asking for $80; cost varies depending on location and the childcare center).
When my child turned 4, we already looked for schools as enrollment for Kindy starts as early as March for next school year. You have to be early as sometimes slots are limited. You would normally enroll your kid to a school near your place or at least within the vicinity/location. There are schools that only accepts students living in the area so if you are aiming to enroll your child to a particular school, you may need to move closer to the school if this is a requirement.
Seasonal Clothing - When we arrived in Sydney, it was autumn and we thought that the jackets/jumpers we have from PH were enough. We were wrong. We had to buy new clothes as we were not prepared for Winter. It is good to buy clothes when it is on sale (normally shops go on sale just right after the season).
Activities - It can be very boring during the first few months especially when you don't know anyone or you don't have any relatives here. My husband wanted to go home after staying for 3 months as there are no activities to look forward to. Try to search for local groups in your area where you will meet new friends and enjoy an activity whether it is a new sport or something that you are already good at (i.e. basketball, volleyball, badminton, tennis, etc.). There are lots of activities you can do when you explore. Try finding some activities in your local area and then go further out once you have the means (i.e. transportation may be needed for some activities so having a car would make it easier but the trains and buses are very reliable so you should be able to do it without the car).