Manhattan prosecutors are poised to push their case against Donald J. Trump into a critical new phase on Tuesday, as they prepare to question a key witness and urge the judge to hold the former president in contempt for attacking witnesses and jurors in the landmark trial.

The case, the first criminal trial of an American president, debuted to a newly seated jury on Monday, as both sides delivered opening statements that offered dueling visions of Mr. Trump and the evidence against him. While a prosecutor accused the former president of orchestrating a “criminal conspiracy and a coverup,” Mr. Trump’s lawyer proclaimed that “President Trump is innocent.”

The prosecution also began questioning its first witness, David Pecker, the former publisher of The National Enquirer, who buried damaging stories about Mr. Trump as he mounted his first campaign for president. Mr. Trump is accused of falsifying records to cover up a sex scandal involving a porn star that could have derailed his campaign.

The flurry of activity set the stage for a weekslong trial that will continue to captivate the political and legal worlds and test the limits of the justice system as Mr. Trump attacks judge and jury alike.

The jury selection process had barely gotten underway last week when prosecutors accused Mr. Trump of violating a gag order intended to prevent any such attacks. The order, imposed by the judge overseeing the case, Juan M. Merchan, bars the former president from attacking witnesses, prosecutors, jurors and court staff, as well as their relatives.

Prosecutors have said Mr. Trump violated it at least 10 times, both in his own statements and by reposting quotes and articles on social media. Prosecutors say those statements and posts have targeted jurors and witnesses — including a key prosecution witness, Michael D. Cohen, Mr. Trump’s former fixer — and risked inspiring violence or harassment by Mr. Trump’s supporters.