WASHINGTON—On the day the Senate moved on long-promised health-care legislation, President Donald Trump signaled his next priority: overhauling the tax code to push corporate rates down and give middle-class taxpayers a break, even if it means some of the wealthiest pay more.

“The people I care most about are the middle-income people in this country, who have gotten screwed,” Mr. Trump told The Wall Street Journal on Tuesday, reiterating that he wants to bring down the corporate tax rate to 15%. “And if there’s upward revision it’s going to be on high-income people.”

Sitting behind his desk in the Oval Office, Mr. Trump hopscotched across a variety of policy and personnel topics over the course of the 45-minute interview.

The president repeated his criticism of Attorney General Jeff Sessions for recusing himself from a probe into Russian meddling in the 2016 election, declining to say the former Alabama senator’s job was safe.

He said his front-runners to be the next chairman of the Federal Reserve board of governors early next year would be the incumbent, Janet Yellen, and Gary Cohn, director of the National Economic Council.