Key moments from Hillary Clinton's official campaign launch speech in New York on Saturday. (AP)

Promising a more hopeful, inclusive America ready to take on the big challenges facing the country, Hillary Rodham Clinton said Saturday that she wants to be the champion the nation needs as well as its first female president.

Speaking before thousands, Clinton blended her work as an advocate for children and families with pledges to give people a fairer bargain on a range of issues, including how taxes are levied, voting conducted and immigration managed. She took shots at her would-be Republican rivals and the very wealthy — a group of which Clinton is a member and on which she will count to help fund a campaign that aims to raise more than $1 billion.

“Prosperity can’t be just for CEOs and hedge-fund managers,” Clinton said in an address on Roosevelt Island, with Wall Street visible over her shoulder. “Democracy can’t be just for billionaires and corporations. Prosperity and democracy are part of your basic bargain, too. You brought our country back. Now it’s time — your time — to secure the gains and move ahead.”

Clinton entered the 2016 race in April but waited two months to frame her populist-flavored agenda and explicitly ask for the nation’s vote. On Saturday, she sought to provide a rationale for her candidacy, saying she is running for “everyone who’s ever been knocked down but refused to be knocked out.”

That line served as both a reference to the economic circumstances of many Americans since the Great Recession as well as to her own disappointing run for the White House in 2008.

Hillary Rodham Clinton greets a crowd of thousands on New York’s Roosevelt Island on Saturday before discussing her agenda. (Melina Mara/The Washington Post)

Clinton was looser than she often appeared during that campaign, smiling and pausing from time to time to absorb the applause from a crowd the campaign estimated at more than 5,000.

Clinton drew her loudest applause with a grinning reference to the many ways she would be a different kind of president. “I may not be the youngest candidate in this race. But I will be the youngest woman president in the history of the United States,” she said. “And the first grandmother, as well.”

She jeeringly referred to what she called the false promise of Republicans that “if we let those at the top pay lower taxes and bend the rules, their success would trickle down to everyone else.”

She did not mention any Republican opponents by name and said nothing about the smaller field of Democratic challengers. But she may have had Sen. Marco Rubio (R-Fla.) in mind when she said that although “there may be some new voices in the presidential Republican choir, they’re all singing the same old song. A song called ‘Yesterday.’ ”

Rubio had made an obvious reference to Clinton in a speech in April announcing his own candidacy, referring to a Democratic candidate “of yesterday.”

In a series of attacks that drew applause from the crowd, Clinton hit Republicans for wanting to repeal the Affordable Care Act, deport immigrants and take away “reproductive-health decisions.” Clinton said Republicans “turn their backs on gay people who love each other.” And on climate change, she said: “Ask many of these candidates about climate change, one of the defining threats of our time, and they’ll say: ‘I’m not a scientist.’ Well, then, why don’t they start listening to those who are?”

As president, Clinton said she would reward businesses for long-term investments. She promised to restore America’s position of being on the cutting edge of innovation, science and research by increasing public as well as private investments. And she pledged to improve preschool options, make college affordable and rebuild decaying infrastructure. Policy details will be rolled out in coming weeks, she said.

Former first lady and secretary of state Hillary Clinton kicked off her 2016 White House campaign with a series of smaller roundtable discussions, aimed at listening and engaging with voters one-on-one. But how will she fare as her campaign ramps up to its next phase: those big, traditional rallies? The Washington Post’s Anne Gearan explains. (Julie Percha/The Washington Post)

Clinton did not dwell on foreign policy, although she noted that over her shoulder stood the United Nations headquarters, where she often represented her country when she was secretary of state.

Her husband, former president Bill Clinton, cheered behind her, as did daughter Chelsea. But it was a tableau arranged to showcase Hillary Clinton and, in particular, her relationship with her mother.

Framed around the story of how Clinton’s late mother, Dorothy Rodham, emerged from a childhood of mistreatment without losing her faith in humanity, the speech laid out how Clinton drew lessons about hope, perseverance and kindness from her mother’s example.

“When I was a girl, she never let me back down from any bully or barrier,” Clinton said. “In her later years, Mom lived with us, and she was still teaching me the same lessons. I’d come home from a hard day at the Senate or the State Department, sit down with her at the small table in our breakfast nook and just let everything pour out. And she would remind me why we keep fighting, even when the odds are long and the opposition is fierce.”

The emphasis on Clinton’s personal story is part of an effort to reshape her image, which has often been of an efficient and, sometimes, chilly policy wonk. She talks with glowing grandmotherly pride about 8-month-old Charlotte at nearly every appearance, though the baby has yet to make an appearance at any of them.

A gauzy, biographical video the campaign released Friday casts Clinton as a lifelong warrior for children and families, showing photos from her 40-year career in advocacy and public service.

“Everyone deserves a chance to live up to his or her God-given potential,” Clinton says in the video. “That’s the dream we share. That’s the fight we must wage.”

The speech’s setting was a nod to both Clinton’s adopted home state, which she represented in the Senate for eight years, and the legacy of President Franklin D. Roosevelt’s ethos that America should be free from want and fear.

“President Roosevelt called on every American to do his or her part, and every American answered,” Clinton said. “It’s America’s basic bargain:  If you do your part, you ought to be able to get ahead. And when everybody does their part, America gets ahead, too.”

That idea of a can-do, all-in-this-together society, with a government that provides and protects, is a startling embrace of the kind of Roosevelt-flavored big government many recent Democrats, including Bill Clinton, have avoided.

That suggests Clinton thinks she can win by appealing to her own party’s most progressive wing as well as to others who feel left behind in an economy where the gap between rich and poor has grown much wider than when her husband was in the White House in the 1990s. It also suggests that Clinton thinks she can overcome her own perceived coziness with Wall Street titans, which has caused ambivalence among progressives.

Clinton immediately flew to Iowa, where she sought to win over a state that was cool to her candidacy in 2008. Clinton enjoys a hefty lead there in early polling, but she faces competition, primarily from Sen. Bernie Sanders (I-Vt.), who has drawn large crowds, and Martin O’Malley, Maryland’s former governor.

In Sioux City, Clinton attended a house party with about 50 supporters while her remarks were simulcast at more than 650 watch parties across the country.

“It’s really not about me. It’s about you, about us, about all the people who deserve a champion,” Clinton said.

She urged watchers to sign up with her campaign online. “Be part of our effort to take back our country,” she said, an odd statement considering the phrase has been a popular Republican rallying cry against President Obama.

Iowa state Rep. Marti Anderson hosted a watch party in Beaverdale, a Des Moines suburb so active for Obama in the 2008 campaign that it was nicknamed “Obamadale.”

A few dozen neighbors gathered in Anderson’s living room to watch Clinton speak. When the candidate said, “I don’t believe we should ever quit on our country,” Anderson, sitting on her piano bench, clapped her hands and cried out, “Yes!”

Rucker reported from Des Moines.