President Obama is to welcome his successor Donald Trump to the White House for talks aimed at smoothing over the transition of power.
It could prove a difficult meeting with Mr Trump having questioned Mr Obama's US citizenship and promising to dismantle some of his key policies.
Mr Obama though has said he is "rooting" for Mr Trump after his shock defeat of Hillary Clinton.
Thousands have taken to the streets of major US cities denouncing Mr Trump.
Mr Obama - who for his part had branded Mr Trump "unfit" for office and campaigned against him - urged all Americans to accept the result of Tuesday's election.
"We are now all rooting for his success in uniting and leading the country," he said.
The defeated Mrs Clinton also told supporters Mr Trump had to be given a "chance to lead".
Despite their calls, protesters gathered in several cities across the country. Many chanted "Not my president".
In New York, thousands marched on Trump Tower, attacking Mr Trump's policies on immigration, gay rights and reproductive rights. Fifteen people were arrested, the New York Times reported.
Protests were largely peaceful but in Oakland, California, some demonstrators smashed shop windows and threw missiles at riot police, who reportedly responded with tear gas.
A mass anti-Trump rally shut down the key 101 freeway in Los Angeles.
In Chicago, crowds blocked the entrance to Trump Tower, chanting: "No Trump, No KKK, No Fascists USA".
In Portland, Oregon, demonstrators temporarily closed an interstate highway.
In Washington DC, protesters held a candlelit vigil. Organiser Ben Wikler told the crowd: "We are here because in these darkest moments, we are not alone".
Demonstrations also took place in Philadelphia, Boston, Seattle and San Francisco, among other cities.
In his victory speech in the early hours of Wednesday, Mr Trump vowed to "bind the wounds of division", after an acrimonious election contest, and to be "president for all Americans".
White House spokesman Josh Earnest has insisted Mr Obama will be sincere about ensuring a smooth handover when he meets Mr Trump, although he added: "I'm not saying it's going to be an easy meeting."
With the Republicans holding a majority in both chambers of the US Congress Mr Trump has an easier path to pass his laws and scrap key Obama schemes like his healthcare reforms.
The president-elect will be accompanied to the White House on Thursday morning (1600 GMT) by his wife, Melania, who will have a meeting with First Lady Michelle Obama in the White House residence.
Mr Obama, who congratulated his successor in a phone call in the early hours of Wednesday, said it was "no secret" that he and Mr Trump had pretty significant differences.
But he added that "we all want what's best for this country" and he was "heartened" by what he heard in Mr Trump's remarks the night before.
Mr Trump's transition team for the 10-week period until inauguration will be led by Chris Christie, the governor of New Jersey.
The president-elect, who has never held elected office, has said his immediate priorities will be restoring the country's infrastructure and doubling its economic growth.
Republican National Committee Chairman Reince Priebus said: "Donald Trump is taking this very seriously," adding that the business mogul's deal-making ability would enable him to quickly "make things happen for the American people".
As president-elect, Mr Trump is entitled to get the same daily intelligence briefing as President Obama, which includes information on covert US operations and other data gathered by America's 17 intelligence agencies.
Mr Trump's team is understood to be focused on quickly filling key national security posts.
But it is not yet clear who will sit in his cabinet or fill senior posts in his administration, such as chief of staff.
There are expected to be roles for Mr Christie, former House Speaker Newt Gingrich and former New York Mayor Rudy Giuliani, another of Mr Trump's closest advisers, who is being linked with the role of attorney general or national security adviser.
After losses overnight as Mr Trump's surprise victory became clear, financial markets rebounded as a feared meltdown failed to materialise.