From small towns in Iowa and New Hampshire to the grand stages of interest groups’ conventions, the 2024 presidential campaign is underway, whether or not Americans are ready.

The past week has brought at least four declared or likely candidates to New Hampshire, three to Iowa and one to South Carolina. Nine addressed the National Rifle Association’s annual forum in Indianapolis, and three attended a Republican donor retreat in Nashville.

The formal choreography of the campaign is falling into place. Last Tuesday, the Democratic National Committee chose Chicago to host its convention next August. On Wednesday, the Republican National Committee, in a surprise to no one, chose Fox News to host the party’s first debate this August.

The declared candidates filed their quarterly fund-raising reports late this week, revealing the first big campaign finance error of the season. The campaign of Nikki Haley, the former South Carolina governor and United Nations ambassador, exaggerated her fund-raising total by more than $2 million by double-counting sums transferred between different committees.

Five major candidates have officially announced campaigns: four Republicans (former President Donald J. Trump, Ms. Haley, former Gov. Asa Hutchinson of Arkansas and Vivek Ramaswamy, a multimillionaire entrepreneur and author) and one Democrat (the self-help author and 2020 candidate Marianne Williamson).

But on the campaign trail, it seems like more.

Senator Tim Scott of South Carolina, who announced an exploratory committee on Wednesday, had a particularly packed week, with trips to Iowa, New Hampshire and South Carolina. A tour of Alex’s Restaurant in Goose Creek, S.C., on Friday had the look and feel of a full-blown campaign stop, with supporters holding signs and the number of reporters rivaling the number of diners.

Mr. Scott talked with voters and restaurant staff before heading outside to take questions from reporters — walking a thin line between being a declared candidate and one in waiting.

“The message is resonating,” he said, underlining his belief that his conservative talking points with religious overtones will appeal to a broad swath of Republican voters. Asked if he had made up his mind about running for president, he said: “I’m getting closer. Without any question.”

He added that he would return to Iowa and New Hampshire in the coming days and had plans to stop in Nevada, another early-voting state.

While Mr. Scott was in South Carolina, Gov. Ron DeSantis of Florida — the top challenger to Mr. Trump in early polls, though not officially in the race — spoke at Liberty University in Virginia and then flew to New Hampshire. Mr. DeSantis addressed a crowd of 500 at a state Republican Party dinner in Manchester.

The event raised $250,000 for the state party, with the party chairman saying Mr. DeSantis had directed his own donors to give an additional $132,000.

After his nearly 40-minute speech, Mr. DeSantis spent just as long methodically working his way through the crowd, visiting all 50 tables for handshakes, backslaps, photos and small talk. “Did you get it?” he asked picture takers. “County chairman for where?”

The low-stakes interactions appeared designed to dispel criticism that Mr. DeSantis was unwilling to engage in the traditional retail campaigning that political activists in early-voting states like Iowa and New Hampshire value. On Saturday, he also stopped by an airport diner.

The governor of New Hampshire, Chris Sununu, was in Nashville, far away from home, testing out his own possible campaign at the Republican National Committee’s private donor retreat. There he spoke at a luncheon on Saturday and implicitly blamed Mr. Trump for the party’s underwhelming performance in the midterm elections. (Data backs him up: A New York Times analysis found that candidates Mr. Trump supported in primaries performed about five percentage points worse than other Republicans did in the general election.)

Mr. Trump was at the retreat, too, casting himself against that evidence as the only candidate who could win a general election. So was his former vice president, Mike Pence, whom Trump supporters declared their desire to hang when they stormed the Capitol on Jan. 6, 2021.

“The old Republican Party is gone, and it’s never coming back,” Mr. Trump said in a speech Saturday, less than two weeks after he was arraigned in New York on 34 felony charges of falsifying business records. “Instead of being the party of the establishment class, we are now the party of the working class, the party of all Americans.”

The evening before, Mr. Pence cast the 2024 election as a fight between “one vision grounded in traditional Republican principles, and another vision that grasps what some think the American people want to hear.” He took repeated but indirect aim at Mr. Trump, noting that in 2022, “candidates that were focused on the past, particularly those focused on relitigating the last election, did not do well.”

On Sunday, Mr. Hutchinson, the former Arkansas governor who announced his campaign this month and was in Iowa a few days ago, partook in another campaign staple: the Sunday morning talk show interview.

Appearing on CBS News’ “Face the Nation,” Mr. Hutchinson gave the usual answer to the question of why he was running — “because we need leadership that brings out the best of America and doesn’t appeal to our worst instincts.” Then the host, Margaret Brennan, pressed him on how he would respond to the country’s bleak parade of mass shootings.

He did not endorse any new federal legislation and expressed skepticism about whether red-flag laws — which allow the removal of guns from people deemed to pose a danger to themselves or to others — protected due process. At the same time, he urged states to make greater use of existing laws that allow the institutionalization of people deemed to pose a danger to themselves or to others.

There has been much less activity across the aisle, where President Biden is inching toward formally declaring a re-election campaign that he has already said was definite. (“We’ll announce it relatively soon,” he said on Friday.)

No one with a large support base has risen to challenge him. But he does have one official competitor, Ms. Williamson, who has been traversing New Hampshire since Friday, hitting Dover, Henniker, Keene, Lancaster and Littleton.

A second challenger, the anti-vaccine activist Robert F. Kennedy Jr., plans to announce his campaign this Wednesday.

The election will be just 566 days away.

Rebecca Davis O’Brien, Shane Goldmacher, Maggie Haberman and Maya King contributed reporting.