Abbott said Monday he hopes to travel to Indonesia for high-level meetings ahead of a summit of Asia-Pacific leaders in the resort island of Bali in early October.
"As for my trip to Indonesia, I want it to happen as soon as possible," Abbott told reporters. "In many respects, this is our most important single relationship."
Abbott led his conservative coalition to a crushing election victory on Sept. 7 partly on a promise to stop an increasing number of asylum seekers from reaching Australian shores aboard Indonesian fishing boats.
His new approach to the problem includes forcing the Australian navy to turn boats back to Indonesia and buying boats from Indonesian fishing villages to prevent them from falling into the hands of people smugglers.
Indonesian officials have criticized both strategies and warned against Australia taking unilateral action.
After Abbott won power, Mahfudz Siddiq, a senior Indonesian lawmaker from Prosperous Justice Party, the largest Muslim-based party and a member of the ruling coalition, was particularly critical of the Australian policy of buying fishing boats.
"His idea is clearly insulting the dignity of Indonesians," he said, "It showed to us that he does not understand diplomacy and threatens our bilateral co-operation with Australia."
Abbott declined to comment on Indonesian attitudes toward the policy.
"I'm determined to get the relationship off to the best possible start and the best way to do that is to indicate to you that I'm not going to conduct discussions with the Indonesians via the media," Abbott told reporters.
Abbott on Monday also named the 42 executive members of his government which included only six women. Only one woman, Foreign Minister Julie Bishop, has been appointed to Abbott's 19-member Cabinet.
Abbott said he was disappointed that Sophie Mirabella, another party stalwart who could have been a minister, would not make it. She appears likely to become the only lawmaker from Abbott's Liberal Party to lose her seat at the election, although vote counting continued.
"Nevertheless, there are some very good and talented women knocking on the door of Cabinet and there are lots of good and talented women knocking on the door of ministries," Abbott said.
Chris Bowen, a senior member of centre-left Labor Party government that was defeated by Abbott after six years in power, said Afghanistan now had more women Cabinet ministers than Australia.
"It's a great shame that we only have one female member of the Cabinet of Australia," Bowen said.
"Can the prime minister-elect look the Australian people in the eye and say not one other female in his entire party room was qualified enough, was meritorious enough, to serve in the Cabinet of Australia in 2013? I find that very disappointing," he added.