WASHINGTON—The Trump administration is moving deliberately to counter what the White House views as years of unbridled Chinese aggression, taking aim at military, political and economic targets in Beijing and signaling a new and potentially much colder era in U.S.-China relations.

In the first 18 months of the administration, ties between the world’s two biggest powers were defined by negotiations over how to restrain North Korea and ways to rebalance trade. Those high-profile endeavors masked White House preparations for...