In his State of the Union address, President Biden laid out a campaign blueprint for the next eight months, assailing Donald J. Trump as a threat to democracy, vowing to protect abortion rights and aiming to reassure voters who are worried that he is too old for the job.

The president’s broadside from the House floor on Thursday, which might end up resembling his remarks this summer at the Democratic National Convention, kicked off what his campaign said would be a furious period of ramp-up activity after months of sluggishness.

Mr. Biden will campaign in the Philadelphia suburbs on Friday afternoon and in Georgia on Saturday, and then travel next week to New Hampshire, Wisconsin and Michigan, his campaign manager, Julie Chávez Rodríguez, said on Friday. Vice President Kamala Harris will appear in Arizona and Nevada, Ms. Chávez Rodríguez said.

“The general election is just starting to crystallize for voters across the country, and we’re taking advantage of the moment to meet them where they are,” she said.

On Friday morning, the Biden campaign also announced a $30 million advertising campaign over the next six weeks. Campaign aides said they expected to hire 350 new staff members and open 100 offices across battleground states in the next month — an announcement that is likely to hush some of the public and private grumbling from allies that the president’s operation has been slow-moving.

The morning after the president framed the election as a choice between him and Mr. Trump on issues of democracy and freedom, his aides sought to compare what they described as their robust operation with what the former president had built.

“Trump’s bleeding cash,” said Jennifer O’Malley Dillon, the president’s campaign chair, who left the White House last month to functionally take over his re-election effort. “He’s really behind in building the infrastructure that you’d expect to be seeing of a former president. He’s really not focused on building new people to his side.”

On a Friday call with reporters, Mr. Biden’s campaign aides did little to hide their frustration with coverage from the political news media and nervous whispers from Democratic allies about polling that shows the president trailing Mr. Trump nationally and in the important battleground states.

The campaign spent $25 million on political advertising in battleground states last fall, an effort that it promoted as the earliest investment on record, only to watch Mr. Biden’s polling numbers worsen as the year progressed.

Mr. Biden’s aides dismissed such public surveys — which largely match private polls conducted by fellow Democrats — and said their internal metrics of political engagement showed Mr. Biden in better shape.

“I think that’s a really important way to look at this in a broader sense,” Ms. O’Malley Dillon said. Polling, she added, is “a faulty measure if it’s a stand-alone.”

The campaign said it would brand Mr. Biden’s increased travel, new advertising and additional staff hires as a “month of action” under the slogan “I’m on Board” — something of an echo of the “I’m With Her” rallying cry that defined Hillary Clinton’s 2016 campaign.

Now the big question for Mr. Biden as he barnstorms the battleground states is how much he can repeat the energetic 67-minute performance he delivered on Thursday.

For a night, Mr. Biden, 81, responded to the doubts about his age that have loomed over his campaign since before it began, with polls showing that his age is among the foremost concerns of voters even within his own party. That political weakness has been accentuated by occasional trips, falls and verbal stumbles that are recirculated on social media with far more velocity than any triumphant clips from his speeches.

He has also been a somewhat intermittent messenger on the topics that serve as the backbone of his case against Mr. Trump: abortion rights and democracy.

He has delivered several speeches about Mr. Trump’s role in the Jan. 6, 2021, attack on the Capitol but has not maintained a sustained effort to make protecting democracy the center of the American political conversation. He also abandoned his 2022 voting rights push about a week after facing resistance in Congress. And as a practicing Catholic, he has long been uncomfortable speaking about abortion.

The speech showed how Mr. Biden is straining to reconstitute the coalition that powered his 2020 victory. That triumph unified left-wing progressives with moderate Republicans alienated by Mr. Trump.

If there was little outreach to the Trump coalition on Thursday night, there was something for each part of the old Biden coalition — including a significant amount of economic policy that aligns with the progressive platform of Senator Bernie Sanders of Vermont. Mr. Biden attacked billionaires and spoke about raising the minimum wage, increasing union power and lowering health care costs.

He did have one cringe-worthy moment when he referred to an undocumented immigrant charged with killing a Georgia woman as “an illegal,” a term far more common among right-wing Republicans than among Democrats. And while he announced new American efforts to deliver aid to Gaza, he did not call for an immediate cease-fire or suggest that he would reduce aid to Israel, as critics of the conflict have demanded that he do.

Though Mr. Biden spoke at more length about the death toll and suffering in Gaza than he has on other occasions, that portion of his speech was not well received by those who have organized recent primary campaigns for “uncommitted” votes to protest his Israel policy. In Michigan, Minnesota and, most recently, Hawaii, growing percentages of Democratic primary voters have made their opposition to the war known.

Still, there is little question that Mr. Biden and his team aim to make the coming months not a referendum on his presidency, but a focus on the perils of what the country would be like if Mr. Trump returned to power. His speech was remarkable for its focus on Mr. Trump — whom Mr. Biden brought up, though not by name, more than a dozen times.

Perhaps no moment illustrated how Mr. Biden wants to frame the 2024 election than when he predicted that the anger that fueled Democratic victories after the Supreme Court overturned a federal right to abortion would continue unabated through November.

“Clearly, those bragging about overturning Roe v. Wade have no clue about the power of women,” Mr. Biden said. “They found out when reproductive freedom was on the ballot, we won in 2022 and 2023, and we’ll win again in 2024.”