WASHINGTON—President-elect Donald Trump and President Barack Obama met for the first time Thursday, with the Republican businessman saying he has “great respect” for a man he’s called “the worst president in the history of the United States” and he plans to seek Mr. Obama’s counsel while in office.

The softening of Mr. Trump’s searing tone from the campaign came as he made the rounds in Washington, where he also met with Republican leaders in Congress who had been cool to his candidacy.

Mr. Obama, who hosted Mr. Trump in the Oval Office for 90 minutes, didn’t have any personal praise for the man he said couldn’t be trusted with the control of the country’s nuclear weapons, but he was jovial during their joint appearance before reporters and pledged to do “everything we can to help you succeed.” He said the two discussed a range of domestic and foreign policy issues and organizational logistics about life in the White House.

Mr. Trump then went to Capitol Hill to meet with Republican leaders House Speaker Paul Ryan (R., Wis.) and Senate Majority Leader Mitch McConnell (R., Ky.).

“As far as I’m concerned it could have gone on for a lot longer,” said Mr. Trump, who was seated next to Mr. Obama in the Oval Office. He called Mr. Obama “a very good man.”