JUDY WOODRUFF: President-elect Donald Trump is back in New York City tonight, after a day of first meetings in Washington. It was the formal launch of the transition of power, and there was no sign of the bitter broadsides that marked the campaign.
John Yang begins our coverage.
PRESIDENT BARACK OBAMA: Well, I just had the opportunity to have an excellent conversation with president-elect Trump.
JOHN YANG: President Obama welcomed Donald Trump to the Oval Office today, just days after he called the Republican dangerous and unqualified, and after Mr. Trump called the Obama record a disaster. So, how did their first face-to-face meeting go?
PRESIDENT BARACK OBAMA: I have been very encouraged by the, I think, interest in president-elect Trump's wanting to work with my team around many of the issues that this great country faces.
We now are going to want to do everything we can to help you succeed, because, if you succeed, then the country succeeds.
DONALD TRUMP (R), President-Elect: I have great respect.
The meeting lasted for almost an hour-and-a-half. And it could have — as far as I'm concerned, it could have gone on for a lot longer. We really — we discussed a lot of different situations, some wonderful and some difficulties.
And I look forward to being with you many, many more times in the future.
JOHN YANG: As reporters shouted questions, Mr. Obama even offered his successor a bit of advice.
PRESIDENT BARACK OBAMA: Here's a good rule. Don't answer any questions when they just start yelling.
DONALD TRUMP: It's always the last one.
JOHN YANG: White House Press Secretary Josh Earnest said the meeting was a little less awkward than some had expected.
JOSH EARNEST, White House Press Secretary: I feel confident in telling you that they didn't resolve all their differences. But I also feel confident in telling you that they didn't try to resolve all their differences.
What they sought to do was to lay the foundation for an effective transition from the Obama presidency to the Trump presidency.
JOHN YANG: Another part of that foundation? Relations with Congress.
When he becomes president, Mr. Trump will preside over a unified government, Republicans in charge here at the White House and on Capitol Hill. The question is how unified those Republicans will be.
Meeting with the president-elect this afternoon, House Speaker Paul Ryan, whose relationship with candidate Trump was sometimes rocky, pledged to work together.
REP. PAUL RYAN, Speaker of the House: We're going to turn that victory into progress for the American people. And we are now talking about how we are going to hit the ground running to make sure that we can get this country turned around and make America great again.
JOHN YANG: Mr. Trump made clear that some of the Republican House's priorities are also his priorities.
DONALD TRUMP: Whether it's health care or immigration, so many different things, we will be working on them very rapidly. And I think we will be putting things up pretty quickly.
JOHN YANG: He also met with Senate Majority Leader Mitch McConnell.
The Trump team is also beginning the arduous task of building an administration. New Jersey Governor Chris Christie, former House Speaker Newt Gingrich and former New York Mayor Rudy Giuliani are being widely mentioned for top positions.
For the "PBS NewsHour," I'm John Yang at the White House.