WASHINGTON—President Donald Trump kicked his “America First” trade policy into high gear Thursday, launching global tariffs on steel and aluminum, while signaling even more aggressive pressure on trading partners to come, especially against China.

In announcing the measures, the president outlined his broader trade agenda, including rewriting existing U.S. pacts and a continuing sweeping investigation of Chinese trade practices, issuing a veiled threat that even bigger penalties are looming against Beijing. “We’re going to cut down the deficits [with China] one way or the other,” he said.

As Mr. Trump ratcheted up rhetorical pressure on China, the U.S.’s largest trading partner, he unveiled metal-industry protections that were considerably softer than opponents had feared a week ago when he first announced they were coming. The president and aides had originally said no countries would be exempt from the 25% steel tariff or 10% aluminum tariff, but on Thursday suggested that a large number of countries could ultimately be spared.

“We’re going to show great flexibility” by considering exempting military allies, Mr. Trump said, and he started by excluding Canada and Mexico immediately. Earlier in the day, he indicated Australia, which he called a “great country” and a “long-term partner” could also eventually be exempted.

The moderated position reflected a week of global lobbying and counter-threats from allies, as well as complaints from U.S. businesses, members of Congress from his own Republican Party, and officials in his own administration.