Many shouted the slogan “Not my president”. Others burned orange-haired effigies of the businessman.
Mr Trump will become the 45th US president after securing a surprise victory over Hillary Clinton.
He is due to meet current White House incumbent Barack Obama for talks aimed at ensuring a smooth transition.
Mr Obama – who 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.
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.
In New York, thousands marched on Trump Tower, voicing their opposition to Mr Trump’s policies on immigration, gay rights and reproductive rights. Fifteen people were arrested, the New York Times reported.
Police earlier erected concrete barriers and other security measures outside the 5th Avenue skyscraper, which is likely to be Mr Trump’s headquarters during the transition to power.
In Los Angeles, a mass anti-Trump rally shut down a key freeway near the city, according to local media.
In Chicago, crowds blocked the entrance to Trump Tower, chanting: “No Trump, No KKK, No Fascists USA” and “Not my president!”
American flags were burned at some protests and in Portland, Oregon, demonstrators temporarily closed an interstate highway. 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.”
The president-elect will be accompanied to the White House on Thursday morning 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”.
Daily briefing by Intelligence Agencies
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 advisors, 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.