Privately, Florida Republicans said he ran a lackluster race for the governorship, and Democrats — who know him mostly from his frequent appearances on Fox News in recent years defending Mr. Trump — expressed skepticism that he would reach across party lines as he has promised.

“Right now a lot of people, and I’m talking about Republicans as well as Democrats, see him as an appendage of Trump,” said State Sen. José Javier Rodríguez, a Democrat who represents Coral Gables and Key Biscayne. “We all know very little about him. His platform during the campaign was virtually nonexistent. So I hope he’s serious when he talks about reaching out to all Floridians.”

Unlike Mr. Scott, who has been on television leveling accusations of “rampant fraud” and urging his opponent to step down, and Mr. Gillum, who launched an oratorical tour of black churches across Florida in the wake of the campaign, Mr. DeSantis has remained largely out of public view since Election Day. He traveled to Tallahassee to set up his transition team and delivered a single statement on video ahead of the litigious recount, noting his lead in the first batch of unofficial results posted a week ago.

He will have to manage a more partisan State House, where Democrats gained some seats this year, and a more evenly divided State Senate, which traditionally has acted as a moderating chamber. Mr. Scott was first elected during the Tea Party wave of 2010 and tried to pass a hard-line immigration bill the following year; the Senate softened his proposal, and the House never took it up.

Mr. DeSantis met on Thursday with State Sen. Bill Galvano, the incoming Senate president, who told reporters on Friday that the governor-elect will inevitably face a learning curve.

“There’s a big difference between what we do here in Tallahassee and what they do in Washington,” he said. “I’m confident he’ll get his footing as we go. But there are some things that I think he will have to discover.”

Tackling the state’s election laws is a possibility, given the weaknesses in the system exposed by the roller coaster of a recount.