Des Moines — Elizabeth Warren may technically be in the "exploratory" phase of a presidential campaign, but her message during her first solo swing through Iowa was clear: "I'm in this fight."

Sen. Elizabeth Warren, D-Mass., greets potential voters during a campaign stop at McCoy's Bar Patio and Grill on January 4, 2019 in Council Bluffs, Iowa. Getty

Throughout five campaign stops across the state, which hosts the nation's first caucuses in 2020, Warren heavily played up her biography as the Oklahoma-born daughter of a janitor who found her life's work in consumer protection and Wall Street accountability. One of the first top-tier contenders to enter what figures to be one of the most crowded fields for Democrats in some time, Warren is aiming to lay down her marker as its resident economic populist and pitch herself as the ultimate fighter for the middle class.

"Why has America's middle class been hollowed out? What is happening to opportunity in this country? Why is the path so rocky?" she said to a crowd of around 500 people in Council Bluffs on Friday evening. "Whatever issue brought you out here tonight," Warren told a crowd of 725 in Des Moines on Saturday night, "I guarantee that it intersects with a Washington that is working for the rich and and the powerful."

In various speeches and question-and-answer sessions, Warren described her early campaign as a fight. "I am dead serious when I say this is the fight of our lives," she said at her first stop. "We're in this fight together," she told an overflow crowd. "It's time to dream big and fight hard," she said, closing out another. "What pulled me in is what is happening to working families across this country. That is the reason why I'm in the fight," she told reporters.

Get Breaking News Delivered to Your Inbox

That fight, of course, also involves President Trump, whom Warren notably made a point not to name during her stops in Iowa, unless prompted by a member of the audience. But it was high in the minds of many voters who came to see Warren and who told CBS that defeating Mr. Trump next year is Priority Number One.

"Oh, it's the most important: get him out," said Cathy Sanchez, a preschool teacher and Warren supporter at an event in Council Bluffs, for example.

In many ways, Warren's trip showcased the different ways Democratic voters are going about looking for a candidate who can take on Mr. Trump. And on that front, there were mixed views on Warren's approach so far.

At an event in Sioux City, one voter questioned Warren's strategy of releasing a DNA test to prove Native American heritage.

"Why did you undergo the DNA testing and give Donald Trump more fodder to be a bully? The woman asked Warren during a question and answer sessions at the Orpheum Theater in Sioux City. Earlier in the week, days after Warren announced her exploratory committee, Mr. Trump tweeted a meme hitting her over the heritage question.

"Yeah, well. I'm glad you asked that question," Warren replied. "I am not person of color. I'm not a citizen of a tribe. Tribal citizenship is difference from ancestry. Tribes, and only tribes, determine tribal citizenship. And I respect that difference."

"I can't stop Donald Trump from what he's going to do. I can't stop him from hurling racial insults. I don't have any power to do that," Warren continued, to which another audience member shouted out, "Yes you do!"

"But what I can do is, I can be in this fight for all of our family," Warren continued. The election is "not about my family, but about the tens of millions of families across this country who just want a level playing field."

While Warren has received backlash within the party for her handling of the DNA test, some other voters who came to see her in Iowa say her overall approach and message competes with the president's.

"She is not going to take any crap from Trump," said Denny Carney, a retiree from Des Moines. "Hillary [Clinton], in the debates, kind of let him run over her. And I don't think she would let him do that. Trump would make a big mistake to go after her."

Others were drawn to her economic pitch, particularly her references to a time when wages matched opportunities. "My daddy ended up as a janitor and I got a chance to become a public school teacher, a college professor, and a U.S. Senator," Warren told the crowd in Des Moines. "Today a minimum wage job will not keep a momma and a baby out of poverty. And that is wrong."

"It reminds us of when we were growing up," said Ed Slattery, a retiree from Des Moines. "It's the Democratic Party getting back to its roots," said his wife Mary, a librarian. Both said they supported Bernie Sanders in the last presidential primary, and voted for Clinton in the general election. Like many of the voters who came to see Warren, the Slatterys were just starting to take a look at the Democratic field and had a ways to go before making decisions about any potential candidates.

"She's true to her word," a Council Bluffs resident named John (who declined to give his last name) told CBS after Warren's event there. He's most concerned about the cost of his prescription medication, which he has to pay nearly $300 for out of pocket. "She's battle-tested, and she's like the tortoise that gets the hares," he said, even though he is also interested in other potential candidates like Joe Biden and Minnesota Sen. Amy Klobuchar.

But even some of those most inclined towards Warren in the primary wondered about general election electability. "She's being portrayed in the right wing media as this wild-eyed liberal from the East Coast," said Bill Shackelford, a retiree from Des Moines. "I don't know if America is ready for her yet, but I hope so," said Sally Carney, a Des Moines resident who plans to back the Massachusetts senator.

For her part, Warren sought to address this potential vulnerability — and implicit comparisons to Clinton's candidacy — as she introduced herself to Iowans.

"All three of my brothers still live in Oklahoma. And one of my three brothers is a Democrat," she said. "We gotta stay focused on what matters to us. And what matters to us is that everybody gets a fighting chance."

"It's not like I have tested this out with some focus groups, or done a bunch of polling," she said in Des Moines and at other stops. "I'm in this fight because it's been the fight of my lifetime."