WASHINGTON—Republican senators chose John Thune of South Dakota as their first new leader in nearly two decades, opting for continuity over calls for radical change heading into populist President-elect Donald Trump’s second term in the White House.

Republicans in the House and Senate elected their leaders Wednesday after securing a trifecta — winning the House, Senate and White House — in last week’s election. In the Senate, Sen. John Thune (R-S.