From Massachusetts, Elizabeth Warren said on "CBS This Morning" that those who voted for her opponent, Republican Sen. Scott Brown, expressed a desire for lawmakers to work together. She says: "I heard that loud and clear."
Obama repeated his campaign slogan of moving "forward" repeatedly in a victory speech early Wednesday in his hometown of Chicago.
"We will disagree, sometimes fiercely, about how to get there," he said. "As it has for more than two centuries, progress will come in fits and starts. It's not always a straight line. It's not always a smooth path. By itself, the recognition that we have common hopes and dreams won't end all the gridlock, or solve all our problems, or substitute for the painstaking work of building consensus, and making the difficult compromises needed to move this country forward. But that common bond is where we must begin."
Former Obama adviser Anita Dunn told "CBS This Morning" that the president made it clear in his acceptance speech that he will be reaching out, and she warned GOP House leaders, representing Ohio, Virginia and Wisconsin, to keep in mind that their voters also wanted to keep Obama.
"Clearly there's a lot of momentum and a lot of incentive for people to work together to really find answers to the challenges," she said.
The vanquished Republican, former Massachusetts Gov. Mitt Romney, tried to set a more conciliatory tone on the way off the stage.
"At a time like this, we can't risk partisan bickering," Romney said after a campaign filled with it. "Our leaders have to reach across the aisle to do the people's work."
Obama won at least 303 electoral votes to 206 for Romney, with 270 needed for victory, and had a near-sweep of the nine most hotly contested states.
But the close breakdown in the popular vote showed Americans' differences over how best to meet the nation's challenges. With more than 90 percent of precincts reporting, the popular vote went 50 percent for Obama to 48.4 percent for Romney, a businessman-turned-politician. Romney had argued that Obama failed to turn around the economy and he said it was time for a new approach that combined lower taxes and a less intrusive government.
Obama's re-election means his signature health care overhaul will endure, as will the Wall Street overhaul enacted after the economic meltdown. The drawdown of troops in Afghanistan will continue apace. With an aging roster of justices, the president probably will have at least one more nomination to the Supreme Court.
A second term is sure to produce turnover in his Cabinet. Treasury Secretary Timothy Geithner has made it clear he wants to leave at the end of Obama's first term but is expected to remain in the post until a successor is confirmed. Secretary of State Hillary Rodham Clinton, Obama's rival for the presidency four years ago, is ready to leave. Defense Secretary Leon Panetta isn't expected to stay on.
Obama won even though exit polls showed that only about 4 in 10 voters thought the economy is getting better, just one-quarter thought they're better off financially than four years ago and a little more than half think the country is on the wrong track.
But even now, four years after George W. Bush left office, voters were more likely to blame Bush than Obama for the fix they're in.
Some Americans were hopeful for progress in Obama's second term.
"He may not have done a great job in my mind but I kinda trust him," Jerry Shul said Wednesday morning in Times Square. "And I feel like he's gonna keep trying and I feel like when people keep trying in you favor things work out. I have faith in him, I have faith he will get with the Republicans and get something done."