WILMINGTON, Del. — Joseph R. Biden Jr. and Senator Kamala Harris made their debut as running mates in a high school gymnasium on Wednesday, pledging to lead the country out of the coronavirus crisis amid an onslaught of attacks from President Trump as the two national tickets went head-to-head for the first time, less than three months before Election Day.The first full day for the newly announced Democratic presidential ticket offered a glimpse of how two once-bitter rivals from opposite coasts and different generations will try to unite Americans around their platform. Projecting warmth toward each other, they sketched out a vision of recovery from the nation’s crises surrounding public health, the economy and racial injustice — challenges that, they argued, Mr. Trump has made worse at every turn with an extraordinarily divisive presidency.“We need more than a victory on Nov. 3,” Ms. Harris said. “We need a mandate that proves that the past few years do not represent who we are or who we aspire to be.”Ms. Harris, a Californian who once served as attorney general of the state, made clear that part of her campaign role would be demonstrating her skills as a prosecutor to build a case against Mr. Trump and Vice President Mike Pence, methodically detailing what she cast as the administration’s failures in combating the coronavirus, reopening the economy and creating conditions under which schools could reopen safely this fall.“Let me tell you, as somebody who has presented my fair share of arguments in court, the case against Donald Trump and Mike Pence is open and shut,” Ms. Harris said.Other contours of Ms. Harris’s role in the campaign also started coming into focus on Wednesday. A Biden adviser described Ms. Harris as well positioned to connect with Black and Latino voters across the country as well as with suburban women, saying that the campaign expected her presence on the ticket to drive turnout in Arizona, Florida and Texas in particular, as well as in communities of color nationally.People familiar with Ms. Harris’s plans said they expected her to be a major presence on the virtual fund-raising circuit, and she and Mr. Biden, the former vice president, held a grass-roots fund-raiser on Wednesday night in an ornate ballroom of the Wilmington hotel where Mr. Biden announced his 1972 Senate candidacy. There, Mr. Biden announced that in the past 24 hours, the campaign had raised $26 million, with 150,000 first-time contributors.More fund-raisers, such as a Bay Area virtual reception, are also planned for Ms. Harris, according to invitations. And her outreach to key Democratic Party constituencies is also underway — an event with members of the Jewish community, for example, is in the works, according to people familiar with the planning. Mr. Trump, who has unleashed sexist attacks on Ms. Harris, called her “a very risky pick” at a news conference as he referred to “horrible things” she had said about Mr. Biden during the primary campaign, suggesting those words would haunt the ticket.“I’m sure that’ll be played back,” Mr. Trump said. “Not necessarily by me, but others. It’ll be played back.”Mr. Trump also defended his administration’s response to the virus, citing the number of tests that have been administered and bragging about the government’s efforts to ramp up production of ventilators to help gravely ill patients.“We have better testing than any country in the world,” he said, adding that “when you look at the job that we’ve done compared to others, we’ve done a great job.”As Mr. Biden and Ms. Harris spoke on Wednesday before a group of socially distanced reporters — not the excited crowd of supporters that would normally greet such an occasion — they each nodded to the symbolism and historic nature of the moment. Wednesday was the third anniversary of the white supremacist rally in Charlottesville, Va., at which Mr. Trump claimed there were “very fine people on both sides,” Mr. Biden noted.“I knew we were in the battle for the soul of the nation,” Mr. Biden said. “That’s when I decided to run. And I’m proud now to have Senator Harris at my side in that battle, because she shares the same intensity I do.”The former vice president, Ms. Harris said, was the only person who had “served alongside the first Black president and has chosen the first Black woman as his running mate.”Ms. Harris, the daughter of immigrants from India and Jamaica, is the first woman of color on a major party’s presidential ticket, and she and Mr. Biden argued that possibilities for American success stories abound despite the challenges that the nation confronts.“Her story’s America’s story,” Mr. Biden said.But they also laid out the staggering toll that the coronavirus crisis has taken on every facet of American life, And they made clear that they will seek to make the election a referendum on Mr. Trump’s handling of the outbreak.“This virus has impacted almost every country, but there’s a reason it has hit America worse than any other advanced nation,” Ms. Harris said. “It’s because of Trump’s failure to take it seriously from the start. His refusal to get testing up and running. His flip-flopping on social distancing and wearing masks. His delusional belief that he knows better than the experts.”The joint appearance, which came a day after Mr. Biden announced his decision, followed a highly public vice-presidential search process. Some of Mr. Biden’s allies made clear their reservations about Ms. Harris, which originated with her searing debate stage attack last summer on Mr. Biden’s record on busing, remarks that struck his team as cynical as she later struggled to articulate her own view on the issue.But in recent weeks, some of the criticism of Ms. Harris from Democrats played out through sexist language around whether she was overly “ambitious,” a dynamic she appeared to nod to when she said she was “mindful of all the heroic and ambitious women before me.”But Ms. Harris and Mr. Biden repeatedly sought on Wednesday to demonstrate that they shared a policy agenda and personal values, with Ms. Harris echoing Mr. Biden’s often-used language about “character,” and demonstrating a fluency with his campaign proposals. They also emphasized the importance of Ms. Harris’s friendship with Beau Biden, Mr. Biden’s son who died in 2015.“Kamala,” Mr. Biden told his running mate, “you’ve been an honorary Biden for quite some time.”Ms. Harris, too, invoked Beau Biden, in a moment loaded with emotion, recalling their frequent phone conversations when they were both state attorneys general.“He really was the best of us,” Ms. Harris said. “And when I would ask him: ‘Where did you get that? Where did this come from?’ — he’d always talk about his dad.”The joint appearance provided a striking reminder of how the pandemic has upended the usual rhythms of a presidential campaign. Four years ago, Hillary Clinton and her new running mate, Senator Tim Kaine of Virginia, made their debut appearance before thousands of people inside an arena in Miami. Mr. Biden, on the other hand, barely strayed from his Wilmington home for his appearance with Ms. Harris.Though scores of onlookers gathered outside Alexis I. duPont High School, the event was closed to the public, depriving it of the typical soundtrack of big campaign speeches. There was no burst of applause, for instance, at the long-awaited moment when Mr. Biden introduced Ms. Harris as “your next vice president of the United States.” (After the event, Mr. Biden told a CNN reporter, “If the science allows us, you’re going to see us campaigning together.”)Still, the current constraints did not interfere with online fund-raising.On Facebook, the Biden campaign sought to keep up the flow of donations on Wednesday, running hundreds of ads prodding supporters to kick in small contributions.The Facebook ads included some broad talking points, saying that Ms. Harris “is a leader in holding the Trump administration accountable” and that “Joe and Kamala are ready to fight for hard-working Americans who have been hurt by Trump and the G.O.P.’s divisive politics.”The Trump team was nearly immediate in its digital response, flooding its multimillion-dollar Facebook campaign with ads calling Ms. Harris a far-left liberal and painting the Biden-Harris ticket as “two of our Nation’s most RADICAL Democrats.”The Trump campaign also turned the Harris announcement into a fund-raising opportunity online, asking for small-dollar donations that it claimed would help air a 30-second, ready-for-TV ad attacking Ms. Harris as a member of the “radical left” and “a phony.”In his own speech, Mr. Biden directly addressed Mr. Trump’s attacks on Ms. Harris, quoting some of the adjectives he had used, like “nasty.”“It’s no surprise, because whining is what Donald Trump does best, better than any president in American history,” Mr. Biden said. “Is anyone surprised Donald Trump has a problem with a strong woman, or strong women across the board? We know that more is to come.”Nick Corasaniti contributed reporting from New York, and Michael D. Shear from Washington.