Troops from Australia and New Zealand were carried in ocean liners, commandeered by the RN, RAN and RNZN. These were steamships, which in peacetime carried cargo and passengers to the rest of the world. Some of these ships were of German ownership, caught in Australian and New Zealand ports at the outbreak of war, but the majority were owned by British, Australian or New Zealand companies. Once a ship was selected to be a transport, they were converted to fit the needs of the role. Space for horses, additional galleys and washhouses were constructed, and existing accommodation rebuilt to fit the needs of the troops. Instead of using bunks, troops aboard ships slept in hammocks, as it allowed more to be fit in to a given space. This could be a rapid conversion - the Demosthenes was converted to carry 1500 troops in 60 hours.

These transports sailed in convoys, as protection from German raiders, and were often heavily escorted. The New Zealand ships would assemble at Wellington, having picked up troops from other parts of the country. They would then proceed to Albany, Western Australia, where the Australian ships had gathered. The first NZ convoy was escorted by two armoured cruisers; HMS Minotaur and the Japanese Ibuki. From Albany, the convoys would proceed to Colombo, on Sri Lanka. For the first convoy, the escorts included both escorts from the NZ convoy, HMAS Melbourne and HMAS Sydney, which would sink the German cruiser Emden. From Colombo, convoys would steam to Aden, then to Suez or Alexandria. Troops would be disembarked at either of these ports. For the Gallipoli campaign, they would re-embark transports at these ports.

Sources:

Sea Transport of the A.I.F, Greville Tregarthen, Naval Transport Board, 1925 - available online from the Australian National Maritime Museum Archives

http://www.flotilla-australia.com/hmnzt.htm