In a White House news conference, Mr Obama also said he "absolutely" has concerns about the president-elect.
But he called on his fellow Democrats to accept the result and "recognise that that is how democracy works".
Mr Obama declined to comment on Mr Trump's pick of controversial conservative figure Steve Bannon as his White House chief strategist.
Speaking of his meeting last week at the White House with Mr Trump, Mr Obama said: "I emphasised to him that, look, in an election like this that was so hotly contested and so divided, gestures matter.
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"I did say to him, as I've said publicly, that because of the nature of the campaign and the bitterness and ferocity of the campaigns, it's really important to try to send some signals of unity, and to reach out to minority groups or women or others that were concerned about the tenor of the campaign."
Mr Obama said of their Oval Office meeting: "We had a very cordial conversation. Do I have concerns? Absolutely."
It was only a week ago that the nation's first black president was describing Mr Trump as "woefully unprepared for the job".
But during last Thursday's White House meeting, Mr Obama said his successor had "expressed a great interest in maintaining our core strategic relationships".
He said this included "strong and robust Nato" partnerships, which he said would convey "enormous continuity" to the world.
Mr Trump on Monday spoke on the phone with Russian President Vladimir Putin, the Kremlin announced, and both agreed their countries' relations were "extremely unsatisfactory".
Mr Trump's appointment of Mr Bannon, editor of the right-wing media outlet Breitbart News, as his senior advisor in the West Wing has stirred controversy.
But Mr Obama told reporters it would "not be fair of me to comment on every appointment that the president-elect starts making".
"It's only been six days," said Mr Obama. "I think it will be important for him to have the room, to staff up.
"There are certain things that make for good soundbites, but don't make for good policy."
He added: "Campaigning is different from governing - I think he recognises that."
Mr Obama predicted "there are going to be certain elements of his temperament that will not serve him well unless he recognises them and corrects them.
"Because when you're a candidate and you say something that is inaccurate or controversial it has less impact than it does when you're president of the United States."
Mr Trump has also named Republican National Committee chairman Reince Priebus as his chief of staff, a relatively uncontroversial pick.
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President Barack Obama has urged Donald Trump to send "some signals of unity" after the US election campaign.