Manchester, N.H. — The race for the Republican presidential nomination headed south Wednesday, one day after a resounding New Hampshire primary win for Mitt Romney that left the rest of the GOP field scrambling to catch the former Massachusetts governor.

South Carolina is next on the primary calendar; voters there head to the polls Jan. 21. Romney’s main rivals are counting on the state’s social conservatism and reputation for brass-knuckle political brawls to slow the former governor’s momentum. The Palmetto State has picked the winner of every GOP nomination fight since 1980.

Romney is hoping a combination of momentum, campaign cash, growing establishment support and a fractured opposition will lead to a victory not only in South Carolina but also in Florida at the end of the month. Romney victories in the first four contests — starting with Iowa last week — could bring the Republican contest to an early conclusion.

“I have a long way to go before I get the nomination,” Romney told CNN Wednesday morning. The other candidates will “find new attacks. [But] I think in the final analysis people want someone who can lead the country back to strength with good jobs and rising incomes, and all these attacks I think will fall entirely flat.”

In his victory speech Tuesday night, Romney sounded like the presumptive Republican nominee.

“Tonight we celebrate. Tomorrow we go back to work,” he told exuberant supporters in New Hampshire. Obama is “a failed president” who puts his faith in government. … We put our faith in the American people.”

Romney, currently ahead in the polls in South Carolina, cruised to victory in New Hampshire Tuesday with nearly 40 percent of the vote. With most precincts reporting, Texas Rep. Ron Paul received 23 percent and former Utah Gov. and U.S. Ambassador Jon Huntsman garnered 17 percent.

Former House Speaker Newt Gingrich and former Pennsylvania Sen. Rick Santorum came in with 10 percent and 9 percent, respectively, and Texas Gov. Rick Perry had 1%.

Romney is the first non-incumbent Republican in modern history to finish first in both Iowa and New Hampshire, the first two contests. He defeated Santorum in the Iowa caucuses by a razor-thin margin of eight votes.

CNN has estimated an overall final tally in New Hampshire of roughly 245,000 votes — the highest number of voters for a GOP presidential primary in state history.

For their part, the other candidates quickly tried to minimize New Hampshire’s importance. Perry noted the fact that he had all but abandoned the state, focusing his time and energy on South Carolina.

“South Carolina is a winner-take-all state,” Perry said on CNN’s “Piers Morgan Tonight.” “Winning here, I can promise you, wipes out the caucus victory and New Hampshire. So if Mitt’s thinking he’s got it in the bag, I think he’ll be in for a great surprise in South Carolina when he shows up here.”

Paul told CNN he expects to raise more money after his second-place finish, and said he is “nibbling at [Romney’s] heels.”

In response to criticism by rivals that his calls for scrapping the Federal Reserve and bringing home American forces from around the world were dangerous, the congressman told supporters Tuesday night: “We are dangerous — to the status quo.”

Huntsman, who had focused virtually all of his time and resources so far on New Hampshire, told his supporters that he’s still “in the hunt.”

Santorum promised Tuesday evening to keep portraying himself as the “true conservative” in the field.

“We can win a huge victory that will rally this country to take on the challenges we have before us,” he asserted.

In an interview with CNN before Tuesday’s results came in, Gingrich acknowledged South Carolina will be vital to his presidential hopes.

“We’re going to go all out to win South Carolina. We think that’s a key state for us,” the former speaker said, describing the race there as a contest between himself — a “Georgia Reagan conservative” — and Romney, “a Massachusetts moderate.”

Earlier in the day, Gingrich argued that if Romney can’t “come close” to 50 percent in New Hampshire, “it’s very unlikely he can sweep the nomination.”

“And I think that gives the party time to take a deep breath, look at his record and begin to realize that maybe this isn’t the right guy to run against Obama,” he told reporters.

Gingrich has been pounding at Romney since Iowa, complaining about a massive negative ad campaign against him by allies of the former Massachusetts governor.

A Gingrich-allied super PAC has already launched its own anti-Romney barrage in South Carolina, and Gingrich and others have honed in on Romney’s years as a financier with Bain Capital, accusing him of getting rich by gutting companies and laying off workers.

Romney will have to answer questions about that in conservative South Carolina, Gingrich said Tuesday.

Asked about the negative ads from the Gingrich camp, Romney told Boston radio station WRKO Tuesday that they “will not help” his rival.

“All I have got to do is keep my head down, keep talking about my message of getting America back to work, my experience in having led two businesses successfully, the Olympics successfully,” Romney said.

Gingrich wasn’t alone in attacking Romney’s business record. In South Carolina, Perry told supporters Romney’s firm “looted” a photo company in Gaffney and a steel company in Georgetown.

“I would suggest they are just vultures,” Perry said. “They are vultures that are sitting out there on the tree limb waiting for the company to get sick, then they swoop in, they eat the carcass, they leave with that and they leave the skeleton.”

Asked about those remarks Wednesday, Romney said the issue has been brought up every time he has run for an office.

“I understand that President Obama is going to try and put free enterprise on trial,” he said. “But, you know, Rick Perry and Newt Gingrich are going to be the witnesses for the prosecution. I’m not worried about that. They can take it as they like, but you saw last night that that approach didn’t work very well for either Rick Perry or Newt Gingrich.”

On Tuesday night, an Obama campaign aide said that despite Romney’s victory in New Hampshire, “the premise of his candidacy began to unravel in the last 48 hours,” referring to the questions over Romney’s business experience.

Romney came under criticism after a speech Monday to the Nashua Chamber of Commerce, when he said he wanted Americans who were unhappy with their health care coverage to be able to switch insurance companies.

“I like being able to fire people who provide services to me,” he said. “You know, if someone doesn’t give me the good service I need, I want to say, ‘You know, I’m going to go get someone else to provide that service to me.'”

The first seven words of that sentence — “I like being able to fire people” — provided an enticing sound bite for opponents to attack.

“Gov. Romney enjoys firing people. I enjoy creating jobs,” Huntsman said at a campaign stop in Concord on Monday.

Romney Communications Director Gail Gitcho said opponents are taking Romney’s remarks out of context — a point on which Gingrich and Paul defended him Tuesday.

However, the attacks fed the image of Romney that his GOP opponents and Democrats have pushed: That he’s a wealthy businessman who can’t connect to average Americans.

Vice President Joe Biden touched on that theme Tuesday night, telling New Hampshire voters via teleconference that Romney’s firing comment was “probably taken a little out of context,” but nonetheless reflected Romney’s true sentiments.

“He thinks it’s more important for the stockholders and the shareholders and the investors and the venture capital guys to do well than for those employees to be part of the bargain,” Biden said.

CNN’s Paul Steinhauser, Alan Silverleib, Dana Bash, Kevin Bohn, Tom Cohen, Jessica Yellin and Rachel Streitfeld contributed to this report.