WASHINGTON, D.C. - Summoned to action by student survivors of the Florida school shooting, hundreds of thousands of teenagers and their supporters rallied in the nation's capital and cities across America on Saturday to press for gun control in one of the biggest youth protests since the Vietnam era. Chanting "Vote them out!" and bearing signs reading "We Are the Change," ''No More Silence" and "Keep NRA Money Out of Politics," protesters packed Pennsylvania Avenue between the Capitol and the White House. 

"If you listen real close, you can hear the people in power shaking," David Hogg, a survivor who has emerged as one of the student leaders of the movement, told the roaring crowd of demonstrators at the March for Our Lives rally in Washington. 

He warned: "We will get rid of these public servants who only care about the gun lobby."

Large rallies with crowds estimated in the tens of thousands in many cases also unfolded in such cities as Boston; New York; Chicago; Houston; Fort Worth, Texas; Minneapolis; and Parkland, Florida, the site of the Feb. 14 attack at Marjory Stoneman Douglas High School that left 17 people dead. 

Protesters complained over and over that they are afraid of getting shot in school and tired of inaction on the part of grown-ups after one mass shooting after another. They called for such measures as a ban on high-capacity magazines and assault-type rifles like the one used by the Florida killer, tighter background checks and school security, and a raising of the age to buy guns.

"I'm really tired of being afraid at school," said Maya McEntyre, a 15-year-old high school freshman from Northville, Michigan, who joined a march by thousands in Detroit. "When I come to school, I don't want to have to look for the nearest exit." 

She added: "I want to get to the problem before it gets to me." 

In Atlanta, Ben Stewart, a 17-year-old senior at Shiloh Hills Christian School in Kennesaw, Georgia, took part in a march in Atlanta to press for what he called "common-sense gun laws." 

"People have been dying since 1999 in Columbine and nothing has changed. People are still dying," Stewart said. "It could be prevented." 

President Donald Trump was in Florida for the weekend. A motorcade took him to his West Palm Beach golf club in the morning. As of early afternoon, he had yet to weigh in on Twitter about the protests.

A demonstrator chants as students and gun control advocates hold the "March for Our Lives" event demanding gun control after recent school shootings at a rally in Washington, U.S., March 24, 2018. \  AARON P. BERNSTEIN/REUTERS

The National Rifle Association went silent on Twitter Saturday morning, in contrast to its reaction to the nationwide school walkouts against gun violence March 14, when it tweeted a photo of an assault rifle and the message "I'll control my own guns, thank you."

About 30 gun-rights supporters staged a counter-demonstration in front of FBI headquarters in Washington, standing quietly with signs such as "Armed Victims Live Longer" and "Stop Violating Civil Rights." Other counter-protests took place nationwide, including in Texas, Boston and Salt Lake City, Utah. 

Organizers of the gun-control rally in the nation's capital hoped their protest would match in numbers and spirit last year's women's march, which far exceeded predictions of 300,000 demonstrators.

"We will continue to fight for our dead friends," Delaney Tarr, another survivor of the Florida tragedy, declared from the stage. The crowd roared with approval as she laid down the students' central demand: a ban on "weapons of war" for all but warriors. 

In Parkland, the police presence was heavy as more than 20,000 people filled a park near the school, chanting slogans such as "Enough is enough" and carrying signs that read "Why do your guns matter more than our lives?" and "Our ballots will stop bullets." 

Gun violence was also fresh for some in the Washington crowd: Ayanne Johnson of Great Mills High in Maryland held a sign declaring, "I March for Jaelynn," honoring Jaelynn Willey, who died Thursday two days after being shot by a classmate at the school. The classmate also died.

Rallying outside the New Hampshire Statehouse in Concord, 17-year-old Leeza Richter said: "Our government will do more to stop us from walking out than it will to stop a gunman from walking in."

Since the bloodshed in Florida, students have tapped into a current of gun control sentiment that has been building for years - yet still faces a powerful foe in the NRA and its supporters.

Organizers hope the passions of the crowds and the under-18 roster of speakers will translate into a tipping point starting with the midterm congressional elections this fall. In addition to pushing for tighter gun laws, the students have been working to register young people to vote.

Polls indicate that public opinion nationwide may be shifting on the issue. A recent CBS News poll found that nearly two-thirds of Americans support stricter laws on gun sales, including an increasing number of Republicans. Sixty-five percent of Americans said as of late February that laws covering the sale of guns should be stricter -- an eight-point increase from December.

Also, a new poll conducted by The Associated Press-NORC Center for Public Affairs Research found that 69 percent of Americans think gun laws in the United States should be tightened. That's up from 61 percent who said the same in October of 2016 and 55 percent when the AP first asked the question in October of 2013. Overall, 90 percent of Democrats, 50 percent of Republicans and 54 percent of gun owners now favor stricter gun control laws. 

But even with claims of historic social momentum on the issue of gun control, the AP poll also found that nearly half of Americans do not expect elected officials to take action. The CBS News poll found that most Americans who support such stricter gun laws -- 59 percent -- are pessimistic about the prospect of the president and Congress enacting them in the near future. A third are optimistic.

CBS News journalists, embedded with survivors of the Parkland, Florida school shooting, take viewers inside the creation of a movement as students turn grief into action in "39 Days", a one-hour documentary to be broadcast March 24, 2018 at 8 p.m. ET/PT on CBS.   

© 2018 CBS Interactive Inc. All Rights Reserved. This material may not be published, broadcast, rewritten, or redistributed. The Associated Press contributed to this report.