DAVOS, Switzerland—President Donald Trump told a gathering of government and business leaders that “ ‘America First’ does not mean America alone,” striking a largely conciliatory tone after what have at times been strained relations between the U.S. and its key global allies and trading partners.

“I am here today to represent the interests of the American people, and to affirm America’s friendship and partnership in building a better world,” Mr. Trump told a packed hall of about 1,600 at the World Economic Forum. He talked...