Abbott, on his first trip outside Australia since taking office three weeks ago, announced a $15 million commitment to a new Australian Center for Indonesia Studies," The Age newspaper reported.
He said the center will be based at Melbourne's Monash University to "build trust and understanding" between the two countries.
Abbott is accompanied on the sensitive mission by his wife Margaret Aitken, Foreign Minister Julie Bishop, Trade and Investment Minister Andrew Robb and 20 business leaders from the financial services, mining, agriculture and healthcare sectors.
With him are Anthony Pratt, chairman of global paper recycling and packaging giant Visy, Catherine Livingstone, chairwoman of mobile phone and Internet provider Telstra, and Nicholas Moore, chief executive officer of Australian international financial group Macquarie.
Indonesian President Susilo Bambang Yudhoyono changed tack on his country's asylum boat policy, which until now has been to approach the issue through regional multinational talks through the Bali Process.
Susilo said Indonesia would make bilateral agreements with Australia to try to solve the people people smuggling problem that remains an issue for both countries, The Age reported.
Indonesia has disliked Abbott's policies, including towing boats back to their departure points and establishing transit and processing centers for asylum seekers on foreign soil.
But after meeting Abbott Monday, Susilo said "Indonesia has striven to overcome this issue, but it would be much better if the cooperation was at the bilateral level."
In return, Abbott said Australia would keep a careful watch on protesters in Australia agitating for independence of the Indonesian territory of Papua.
Indonesia's provinces of West Papua and Papua together are about the size of Spain and occupy the western half of the island of Papua, off Australia's northern coast. The independent country of Papua New Guinea occupies the eastern half.
Australia maintains "total respect for Indonesia's sovereignty, a total respect for Indonesia's territorial integrity," Abbott said.
"The government of Australia takes a very dim view ... of anyone seeking to use our country as a platform for grandstanding against Indonesia. We will do everything that we possibly can to discourage this and prevent this," he said.
Details of any changes in people smuggling and asylum boat transfer policies will have to be thrashed out between Australian Immigration Minister Scott Morrison and his Indonesian counterpart, Djoko Suyanto.
Australia has been taking in thousands of refugees each year.
During his election campaign, Abbott, leader of the Liberal National party, promised a tougher line on the people smugglers' boats, many of which are un-seaworthy.
The Labor party, which lost to the Liberal party, was sending asylum seekers directly to detention camps in Papua New Guinea and the Micronesian island nation of Nauru, with no hope of entering Australia under refugee status.
The Liberal plan includes a senior military commander as head of a force to go after people smugglers and to coordinate boat arrival operations.
Abbott's Operation Sovereign Borders plan is for Australia's 12 agencies involved in border protection to come under the command of a single three-star military commander.
Many of the overcrowded rickety smuggling vessels leave Indonesia and Malaysia and break up in rough seas in the Indian Ocean.
Australian officials dismissed claims last week that it took 24 hours for authorities to respond to Indonesian asylum seekers whose boat capsized.
The boat floundered Thursday off Indonesia's south Java coast, leaving as many as 50 people, including 30 children, dead or missing, the Australian Broadcasting Corp. reported.
"I called the Australian government, like 12 times [on Thursday]. I told them we have 35 children," one survivor said, adding that he told authorities that, "We don't want to go to Australia, just take us out of the water. We don't want to die. It's our mistake, not the children's mistake."
Morrison issued two statements denying the delay, saying Australia didn't receive a call for help until Friday, at which point authorities immediately sent out search-and-rescue vessels.
Survivors of the boat sinking also alleged members of the Indonesia military were the ones who brought the asylum seekers to the boat to be smuggled into Australia.
"The army took us," another survivor said. "The army was driving the cars."
Abbott returns to Australia Tuesday.