There were protests against his election across the US on Wednesday night, with crowds in New York chanting "not my president" outside his Manhattan residence, Trump Tower.
The victor will head to the White House on Thursday as president-elect for talks with Mr Obama, who will hand over power in January.
The controversial tycoon is also expected to have a conversation with Theresa May "at the earliest opportunity" as the world waits on tenterhooks for signs of what a Trump presidency means for global politics.
Downing Street confirmed the Prime Minister is preparing for a phone call with Mr Trump as the "special relationship" between the UK and US begins a new chapter.
Mrs May was at pains not to endorse either Mr Trump or his defeated Democratic rival Hillary Clinton during the bitter campaign, although she has previously described the president-elect's call to ban Muslims from entering the US "divisive, unhelpful and wrong"."
Britons will be keen for any indication of what Mr Trump's victory, which sparked horror in some quarters of British politics, means for the future of the transatlantic alliance.
The Republican regularly evoked Britain's shock vote to leave the European Union during his campaign for the presidency, insisting his victory would be like "Brexit plus plus plus".
Although he has run a generally anti-global trade campaign, Mr Trump suggested in May that Britain would be favoured for any future trade deals, saying: "You'd certainly not be back of the queue, that I can tell you."
Mrs May on Wednesday wrote to Mr Trump to congratulate him on his victory, declaring that the UK and America will remain "strong and close partners on trade, security and defence".
The Republican has already spoken to the Israeli prime minister Benjamin Netanyahu, but the PM is likely to be near the front of the queue of other world leaders despite a potential realignment of America's global stance.
Many will be waiting to see the direction of Mr Trump's first conversation with Vladimir Putin, who he as repeatedly praised.
The Russian president has already sent Mr Trump a telegram of congratulation and expressed "his hope to work together for removing Russian-American relations from their crisis state".
As part of the transition of power, the White House said classified intelligence materials, including the president's daily brief, are now being made available to Mr Trump, as it was a courtesy extended to Mr Obama and his team by George W Bush.
White House spokesman Josh Earnest has also insisted Mr Obama would 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."
Despite calls for unity from Mrs Clinton and Mr Obama, Mr Trump's win sparked protests in several US states.
The former reality TV star's comments about Muslims, his calls to build a wall to keep Mexican migrants out, and secretly taped recordings of him joking about sexual assault are among the controversies that have attracted widespread and deep opposition.
After his win, demonstrators in Oakland, California smashed windows and set bins on fire. In Portland, Oregon, reportedly hundreds of anti-Trump protesters burned American flags and chanted "that's not my president" while blocking traffic and causing rail delays.
Protests were also seen in Pennsylvania, Seattle, Phoenix and elsewhere in California.
After defying almost all pollsters' predictions, Mr Trump attempted reassure his opponents, promising to be a "president for all Americans" in a move that steadied financial markets.
Observers will also be looking for indications about the future in who he appoints to his cabinet and how he fills a vacancy in the Supreme Court.
Former New York mayor Rudy Giuliani, former House of Representatives speaker Newt Gingrich, and New Jersey governor Chris Christie are among those tipped for senior posts.