In the coming months, Obama will have to decide where to be ambitious, where to be cautious, and where to buy time.
He draws political strength from his surprisingly easy re-election in a bad economy. It's partly offset, however, by Republicans' continued control of the House, plus their filibuster powers in the Senate.
Some of the big issues awaiting the president's decisions are familiar, long-simmering problems. They include immigration and the need for a tenable balance between taxes, spending and borrowing.
Another issue, gun control, jumped to the national agenda's top tier this month following the massacre of first-graders and teachers in a Connecticut school. And the issue of climate change remains unresolved.
Veteran politicians and presidential historians say it's almost impossible for Obama to "go big" on all these issues. Indeed, it might prove difficult to go big on even one. While some counsel caution, others urge the president to be as bold and ambitious as possible.
"Americans are yearning for leadership," said Gil Troy, a presidential scholar at McGill University.
As a president dealing with policy, he said, Obama has generally failed to give "that visionary, powerful address that we came to know and love and expect in the 2008 campaign."
Rather than let Congress take the lead on big issues, as it did in drafting the 2009 health care overhaul, Obama should be more forceful in pushing new legislation or using his executive powers to bypass Congress where possible, Troy said.
"The gun control issue is a major opportunity for Obama to make his mark on history -- and solve a problem that has frustrated Democrats for decades," he added.
Other presidential historians, however, think Obama is severely constrained by political realities. They say he will have to carefully pick and choose which goals to emphasize in his second four years.
"I see Obama as almost uniquely handcuffed by circumstances," said John Baick of Western New England University. The number of big, unresolved problems facing the nation, coupled with a deeply divided public and Congress, he said, leave Obama with fewer viable options than most presidents have enjoyed.
At best, Baick said, the U.S. government "is a gigantic cruise liner, and the most he can do is keep us from hitting icebergs."
For instance, Baick said, "if he goes big on gun control, then it's 1994 all over again."
Then-President Bill Clinton pushed an assault weapons ban through the Democratic-led Congress that year, prompting fierce pushback from gun-rights groups. Clinton later would credit the NRA with shifting the House majority to the GOP for the first time in 40 years. However, other factors -- including a House bank scandal -- played big roles, too.
Paul Rego, a political scientist at Messiah College in Grantham, Penn., largely agrees with Baick.
"While President Obama does not face the same cataclysmic events that Abraham Lincoln faced, or that FDR encountered in the form of the Great Depression and World War II, his challenges are many and significant," Rego said in an email.
He said Obama "faces a hurdle that neither Lincoln nor Roosevelt had to overcome during the tumultuous years of their respective presidencies: divided government." Today's Democrats and Republicans differ so sharply about government's proper role, Rego said. He said that Obama's job "is actually harder than that of his most illustrious predecessors."
Politicians of all stripes say Obama's first priority is to resolve the deep partisan divide over tax-and-spending issues, exemplified by repeated impasses over two years that led to this week's showdown on the "fiscal cliff."
An even higher-risk conflict may arise in a few months. Congress again must either raise the federal debt ceiling or see the government default on its loans.
Beyond that, lawmakers and interest groups are watching for signs of how hard Obama might push to restrict firearms and expand illegal immigrants' rights.
Obama said last Wednesday that gun control will be a central issue in his second term. "I will use all the powers of this office to help advance efforts aimed at preventing more tragedies like this," he said of the Sandy Hook Elementary School mass killings.
The president named an interagency task force to recommend anti-violence legislation within weeks. The strategy gives him room to distance himself somewhat from its recommendations if he wants, even though he named Vice President Joe Biden to chair the panel.
Americans' affinity for firearms runs deep, and many political activists think Obama could have more sweeping success with immigration changes.
He won a big majority of Hispanics' votes in both his elections. The trend alarms Republican strategists, who fear their party won't win another presidential election until it repairs its bad relations with Latinos.
With Democrats and Republicans increasingly aware of Hispanics' growing political clout, "this might be an historic opportunity," Troy said.
Chris Dolan, a political scientist at Lebanon Valley College in Pennsylvania, agrees. He said he expects Obama to be "incredibly ambitious on comprehensive immigration reform."
The effort, Dolan said, could "build a lasting Democratic support group. You can't do that with gun control."
Still, opposition to granting citizenship to illegal immigrants runs deep in many circles, especially the Republican Party's base. Bids for "comprehensive immigration reform" have gone nowhere in Congress in recent years.
Several advocacy groups want Obama to make the most of his executive powers to enact measures that don't require congressional action.
The Migration Policy Institute earlier this year made several suggestions regarding immigrants. They included "establishing uniform enforcement priorities," defining "what constitutes effective border control," and "allowing applicants for immigrant visas to file in the United States."
Now that Obama has won re-election, however, the advocacy group wants him instead to push a broader agenda through Congress.
"With the issue teed up for possible action," said Doris Meissner, a former commissioner at the U.S. Immigration and Naturalization Service, antagonizing congressional Republicans with executive actions "would not be politically smart."
The political climate for sweeping immigration changes "is significantly better," Meissner said, "but that does not mean it will happen."
Even with a full plate of challenges and a hostile party controlling the House, she said, "I think Obama absolutely has to go big on immigration."
The White House has declined to detail the president's plans for a second-term agenda. Once the deficit-spending problems known as the "fiscal cliff" are addressed, said White House spokeswoman Jamie Smith, "President Obama looks forward to working on a number of issues that are critical to our future, from immigration to energy, to education and national security direction."