They were generally led by brigadiers and generals who were successful in World War One, and their familiarity with Europe meant Australians fought well against Germany as the tactics Australian forces were using in 1918 were similar to those applicable to 1941-3. Australian strength in infantry operations applied well in the desert.

However, against Japan they were much less successful and in 1941-2 Australian forces performed poorly against Japan (eg 8th Division, Lark Force) in large part because of a racist underestimation of Japanese forces that was common to the Allies.

In New Guinea, specifically Milne Bay and Kokoda, they re-learnt jungle warfare against Japanese forces of limited size and very limited logistical capacity.

By 1943 they were sidelined because MacArthur wanted all the glory for Americans and was disparaging of Australian forces.

Tactically, the Australian army used British methods based on small unit fire and movement, one unit covering the other to advance or retreat. They tended to 'raid' aggressively to keep the enemy at arm's length, not stay stuck in their positions.

Many Australians fought in Bomber Command as well.

To expand upon PantsTime's remarks: Australia was generally unprepared for another major conflict, belatedly following Britain's lead in rearming in the late-1930s. Upon the outbreak of war, the country had around three and a half thousand permanent soldiers and a reserve estimated at around eighty thousand. In 1939-40, prior to the fall of France, most people expected events to be somewhat similar to the Great War, in particular a prolonged conflict in western Europe. This would have allowed Australia, and the other Dominions, to rearm and train men relatively slowly over a prolonged period, but of course did not transpire. The first Australian and New Zealand divisions sent to support Britain were kept in North Africa, where they had been training, due to the crisis in France in the spring and summer of 1940.

For operations, Australians were expected to be fighting alongside the British and the British Commonwealth Visiting Forces Act (1933 - notably two years after the Statute of Westminster) was supposed to govern the rules of cooperation and command for the Commonwealth (Britain and the Dominions - Australia, New Zealand, Canada and South Africa). In North Africa, however, Australia and New Zealand notably chose to give their top commanders a new charter of rights, to better protect the welfare of their troops while cooperating under overall British command. This was in light of WWI ideas of British officer incompetency, relatively new formal independence for the Dominions and the military legacy of the Great War. Crucially these charters allowed Australian commanders to withhold their units from action and appeal directly to their own governments if they thought operations were unduly hazardous for their men.

Regarding weaponry, particularly for Australian overseas forces, the majority was supplied by the UK in the first half of the war. The composition of infantry divisions differed between Australia and the other Commonwealth members upon the outbreak of war, but the Dominions largely followed Britain's lead on this and standardised in the early months of the war. In all, Australia sent 3 divisions to the North African theatre to fight within Britain's Middle East Command.

In Australian operations with Middle East Command (which included Australian actions in Libya, Greece, Crete, Syria and Egypt) the main influences on Australian fighting techniques were the overall British Commander Auchinleck and the head of Australia's forces, and Auchinleck's second-in-command, Blamey.

The debates these two had over doctrine and the use of forces were affected by national loyalties and war experience. Auchinleck was a product of the British Indian Army structure and Blamey was at times resentful of overall British control, and agreed with his government that it was desirable for Australia's three divisions to fight as a single corps when possible (which was scarcely possible given the range of operations being undertaken by the command). Blamey claimed being the head of a Dominion Army and simultaneously second to Auchinleck was a dual role that required a Machiavellian touch.

Auchinleck favoured using brigades and brigade groups in operations in North Africa, believing that they offered the greatest mobility to his forces due to their smaller size. Others countered that this parcelled out the strength of Commonwealth forces into segments too small for decisive success. Auchinleck nevertheless felt that Blamey (and his government) was too caught up with the Great War experience to understand the new levels of mobility required in desert operations. Other notable influences of Australian forces were Auchinleck's predecessor and successor, Wavell and Montgomery respectively.

A classic example of Australians not simply holding fixed positions was in defending Tobruk - their defence was set up to quickly smother break throughs of their lines, rather than merely prevent any breakthroughs occurring, which was notoriously difficult in the desert war.