Once you get the air groups to Hawaii, you need to develop all the little intermediate bases so you can "hop" to the South Pacific and Australia. Johnston Is., Christmas Is., Canton Is, Sauva, ect., should be expanded with base size development, engineers and avaition support (Pay attention to troop size limitiations on some of these bases). If I recall, B-25 Mitchell's can hop Hawaii -> Christmas -> Souva -> Australia (Brisbane). You can do one hop per day, so you can deploy them to Australia in a week. They may develop fatigue or damage airframes on the way, hence the avaition support.
Fighters you usually have to haul by ship to Australia. Later on, when the newer allied fighters arrive, you can hop some of them all the way to Australia. The key is to enable drop tanks, yeah I had a duh moment when I finally figured it out, to extend the transfer range of the fighters. P-38's and P-47's can be hopped in that manner. Later on you can move fighters into the Central Pacific also as required.
Just a note, early on, especially in PBEM, do not move anything extra beyond your requirements to these bases unless you move in infantry and artillery to properly defend them. The line of communication between Hawaii and Australia is a natural target for the Japanese player and they may come calling. I would not risk a CV battle over even Christmas Is. in 1942. You can make him work to take them if he wants, but that is all I would risk.