Australia had a a fairly large army, but the bulk of them were known as Territorial Army units. General Blamey told Douglas MacArthur that they were like Koala Bears, you can't shoot them and you can't export them. There were four divisions of volunteer Australian army troops on December 7th 1941. They were younger than most of the Territorial troops, and were legally able to fight overseas. Three of these divisions were in North Africa and one was in Malaysia. The division in Malaysian was lost when Singapore surrendered on February 25th 1942, but the three division in North Africa were returned to Australia, They were sent to Papua New Guinea and fought the Japanese Army hard in 1942 and 1943. They prevented the Japanese Army from capturing Port Moresby, the only large town in New Guinea and pushed them back over the Owen Stanley mountain range. They also captured Japanese bases of Gona by December 9th, 1942. This was nearly a month sooner than the US Army captured the nearby base of Buna on January 3rd 1943. The Australians provided most of the troops that captured the large Japanese base at Lae in August 1943. In 1944, they mopped up Japanese forces in New Guinea and participated in the closing stages of the battle of Bouganville. In 1945, they were used to capture some targets in Indonesia, like Balakapan and Timor.
These units were the Australian 6th, 7th and 9th divisions. They were very important in stopping the Japanese army in New Guinea in 1942 and early 1943. After that, MacArthur tended to use Australian troops to contain Japanese troops that had been by-passed in 1944 and early 1945. He was also rather stingy with providing the amphibious sea-lift needed to liberate all of the Indonesia targets the Australians wanted to attack, or invade Malaysia and the Malaysian controlled portions of Borneo. General Blamey and Douglas MacArthur worked well together in the dark days of 1942 and into 1943, but their relationship became strained as Blamey felt his Australian soldiers were being shunted off to more secondary tasks during the final two years of the war.
Sources: "Old Soldiers Never Die: the life of Douglas MacArthur" by Geoffrey Perret
"The Pacific War Atlas 1941-1945" by David Smurthwaite