In May, a blue and white 747 emblazoned with the presidential seal cruised over Saudi Arabia and Jordan before entering Israeli airspace, making a short but historic journey from Riyadh to Tel Aviv.

That flight, carrying President Donald Trump on his first foreign trip, was remarkable not just because it was the first direct presidential flight between Saudi Arabia and Israel. It also was a potent symbol to the world that Mr. Trump had his own ideas about America’s best friends and enemies.

Gone were the days of visiting Mexico and Canada first—something every president since Reagan (and nearly all of his predecessors) have elected to do. To the contrary, Mr. Trump has soured both of those relationships, repeatedly threatening to withdraw from the North American Free Trade Agreement and even telling The Wall Street Journal last week that he would use negotiations on the pact to pay for a controversial wall on the U.S.-Mexico border.

In a bid to correct what he views as the faults of his predecessor Barack Obama’s foreign policy, Mr. Trump has reshuffled the deck of American relationships, elevating Gulf Arab leaders, alienating Europeans and eschewing some of the tough talk typically reserved for the heads of China and Russia, diplomats, former officials and analysts said.

A White House official, however, noted that Mr. Trump has formed improbable friendships with Canada’s Prime Minister Justin Trudeau and French President Emmanuel Macron. “They don’t look like Trump types,” the official said.