WASHINGTON — A new group dedicated to promoting President Biden’s ambitious agenda is beginning a multimillion-dollar ad campaign trumpeting his Covid recovery package and infrastructure proposal while contrasting Mr. Biden’s low-key style with his bombastic predecessor’s.Building Back Together, a progressive organization run by Biden allies, will air minute-long television commercials next week in Pennsylvania, Nevada, Georgia and Wisconsin that highlight the president’s response to the coronavirus and his wide-ranging economic plans. The group is planning to spend over $3 million on a monthlong effort, including a shorter advertisement that will appear on digital platforms in the same four states and in North Carolina.Both spots differentiate Mr. Biden’s approach from that of former President Donald J. Trump.“You won’t hear him yelling or sending angry tweets, because for Joe Biden, actions speak louder,” says a narrator in the television commercial.The shorter digital advertisement concludes, “No drama, just results.”The strategy illustrates how determined Democrats are to effectively keep running against Mr. Trump. He may be out of the White House, and barred from sending angry or insulting tweets, but his approval ratings have fallen even lower since he left the presidency, and he remains the best foil for Mr. Biden, who has kept an unusually low profile for a new president.Mr. Biden, meanwhile, is drawing solid if not spectacular early marks, a reflection of the country’s deep polarization.So as he turns to an expansive, and expensive, menu of domestic proposals aimed at lifting the economy, combating poverty and addressing climate change, his supporters are hoping to retain support from the voters who helped lift him to victory last year in part by reminding them of Mr. Biden’s predecessor.“The message is simple: Chaos is out, competence is in, and help is here for Americans,” said Stephanie Cutter, an adviser to Building Back Together who is close to Mr. Biden and top West Wing officials.The group, whose formation was first reported in February, is going on the air in vote-rich and costly markets: Las Vegas, Atlanta, Philadelphia and Milwaukee, as well as Scranton, Pa., Mr. Biden’s childhood home. The group has drawn some scrutiny because it has said it will not disclose the identity of its donors.The ads are aimed at winning over people of color, upscale white suburbanites and the smaller group of working-class whites who moved from supporting Mr. Trump in 2016 to backing Mr. Biden in 2020. Building Back Together is particularly focused on retaining the sort of independent or even Republican-leaning voters who backed Mr. Biden but might have voted for G.O.P. candidates farther down the ballot.The goal for this and future ad blitzes, officials say, is to try to cement the president’s new coalition by reminding them of what they may have disliked about Mr. Trump and by pitching Mr. Biden’s agenda. They hope that by mixing television and digital they will reach voters across platforms and throughout the day.In addition to being crucial presidential battlegrounds, Georgia, Nevada, Pennsylvania and Wisconsin also have some of the most significant races for Senate and governor seats next year.This new effort, while chiefly aimed at promoting Mr. Biden, could also help Democratic candidates in those states whose fortunes in the midterms will be tied in large part to the president’s popularity. Many in the party, including Mr. Biden himself earlier this year, have said that former President Barack Obama did not do enough to highlight his early agenda and paid a price for it in the 2010 midterm elections.This is the group’s first advertising campaign, but the organization intends to be active as the main outside group for Mr. Biden, at least through next year’s midterm election. The name is taken from the president’s campaign slogan, which has become shorthand for his post-Covid economic proposals.