The ceremony included US Marine band performances of the national anthem of both countries and a 19-gun salute and inspection of troops.
Prime minister Scott Morrison will meet with Mr Trump on Friday ahead of a state dinner later that night.
Trump held his first state visit for French President Emmanuel Macron past year.
The White House says the Trumps gave the Morrisons a custom-made leather photo album hand-inscribed with 23-karat gold and a Tiffany & Co. pewter bowl engraved with the presidential seal and the signatures of both Trumps.
He also said the world is a better place with the United States, and that the two nations "see the world through the same lens".
According to the first lady's office, which planned the dinner, the colors for the visit are green and gold, Australia's national colors.
The full guest list and first lady's gown details will be released at the beginning of the dinner, the White House said. The dessert: lady apple tart with calvados ice cream.
Dinner decorations were to feature more than 2,500 roses from California in shades of yellow, as well as Australia's national flower, the golden wattle, while the garden was to be decorated with white and yellow roses. The U.S. military is providing the entertainment.
Trumps host Australian prime minister for rare state dinner in Rose Garden
Mr Morrison said the visit would allow Australia to lay the foundation for "another 100 years" of mateship.
"There are many larger, I suppose, more powerful friends that America has, but they know that they do not have a more sure and steadfast friend than Australia", he told reporters who accompanied him to Washington.
"I'm looking forward to celebrating tonight", said Rep. Mark Meadows, R-N.C.
Trump gave Morrison a model of the USS Canberra, which is now being built at an Alabama shipyard, and a framed map of the Battle of the Coral Sea, a World War II battle in which the USA and Australian naval and air forces fought together against the Japanese.
Beyond socialising, the leaders have planned talks on military, intelligence and economic issues, focusing on the Indo-Pacific region.
"The United States has also benefited significantly from our trade, having enjoyed a trade surplus with Australia since the Truman administration", he said. On Friday, the two will appear together before the press.
Morrison said in his speech that Australia shares the same "truths and traditions" as the United States and sees the world through the same lens.
Australian pro golfer Greg Norman, a friend and occasional golf partner of Trump's, is among the invited guests. Roy Blunt of Missouri; and Lachlan Murdoch, CEO of Fox Corp., the parent of Fox News Channel, one of the president's favored cable TV networks. Plan B: take the White House state dinner party indoors in the unlikely event that raindrops start to fall.
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