President Barack Obama faces a lengthy and familiar set of challenges after riding a wave of support from moderates, women and minorities to a re-election victory over Republican challenger Mitt Romney.

With final results from Tuesday's vote still being counted, Obama had won at least 303 electoral votes and led Romney by more than 1 million votes overall to claim a surprisingly solid victory in a race that polls and pundits had predicted would be tighter.

In addition to giving Obama and Vice President Joe Biden another four-year term, voters elected another Republican majority in the U.S. House and preserved the narrow Democratic majority in the Senate.

Now Obama and the new Congress that will look a lot like the old one will face fresh pressure to legislate a comprehensive deficit reduction deal that has been stymied so far by intransigence on the issue of tax reform.

In the short term, that means dealing with the so-called fiscal cliff -- a combination of tax hikes and mandated across-the-board spending cuts set to kick in at year's end.

Failure to address the nation's chronic deficits and debt is considered a drag on economic growth and job creation. The lack of confidence in political will to find solutions contributed to a first-ever downgrade of the U.S. credit rating in August 2011.

Obama used his victory speech to euphoric supporters to warn of further fights and frustrations in their shared quest to restore equal opportunity for all.

"We are an American family and we rise or fall together as one nation," Obama said to prolonged cheers. "Tonight, in this election, you, the American people, reminded us that while our road has been hard, while our journey has been long, we have picked ourselves up, we have fought our way back, and we know in our hearts that for the United States of America, the best is yet to come."

He emphasized that the political arguments that come with democracy were a necessary part of the process.

"We will disagree, sometimes fiercely," Obama said, noting that "progress will come in fits and starts" and the victory Tuesday night "won't end all the gridlock."

Foreshadowing hard decisions ahead, the president said blind optimism and wishful idealism "can't substitute for the need to make difficult compromises to move forward."

House Speaker John Boehner, who backed away from a possible deficit reduction "grand bargain" with Obama last year over the president's demand for some tax hikes as part of the deal, immediately signaled a continued hard line.

"The American people re-elected the president, and re-elected our majority in the House," Boehner, R-Ohio, said in a statement. "If there is a mandate, it is a mandate for both parties to find common ground and take steps together to help our economy grow and create jobs, which is critical to solving our debt."

Reassess message and tactics

Other Republican leaders said the generally bad election for the GOP, which included losing the presidency and failing to win a Senate majority despite Democrats having to defend twice as many seats, meant the party needed to reassess its message and tactics.

"It's clear that with our losses in the presidential race and a number of key Senate races, we have a period of reflection and recalibration ahead for the Republican Party," said Sen. John Cornyn of Texas, the chairman of the National Republican Senatorial Committee. "While some will want to blame one wing of the party over the other, the reality is candidates from all corners of our GOP lost tonight. Clearly we have work to do in the weeks and months ahead."

Analysts agreed, arguing that the tea party movement helped defeated moderates such as veteran Sen. Richard Lugar of Indiana in the primaries, only to see the social conservative who ousted him -- State Treasurer Richard Mourdock -- lose in Tuesday's general election.

"It's not about geography anymore with the Republican Party," said Margaret Hoover, a GOP strategist and CNN contributor. "It's about demographics, and we've got to start thinking about growing the party."

David Gergen, CNN's senior political analyst, said the Republican Party must move its foundation back toward the political center from the increasingly right-wing positions pushed by tea party and social conservatives.

"It seems to me that the lesson has to be clear to Republicans that they have to adjust," Gergen said. "They've gotten too far out."