WASHINGTON (CNN) -

President Obama began his second term with a "keeping it real" moment courtesy of youngest daughter Sasha.

"You didn't mess up," she told the president after the oath was administered by Chief Justice John Roberts in the White House Blue Room.

It all went off without a hitch compared to four years ago when a redo was required after flubs during the swearing-in ceremony at the Capitol raised doubts about its legitimacy.

With a sense of relief on Sunday the president declared, "I did it."

But over the next several months, those words will be much more difficult as the president pushes Congress to embrace his priorities, from gun control to immigration reform to raising the debt ceiling.

"We've had a pendulum of instability, arguing, fighting," said CNN contributor and Republican strategist Ana Navarro.

Compromise was Washington's needle in a haystack, discovered often in the 11th hour.

But now there's a tinge of optimism as the president gets to work in his second and final term.

"I think we'll see it (pendulum) come back to the middle sooner rather than later," Navarro said.

Gun control

The first big test is shaping up to be an issue that never came up on the campaign trail and wasn't even on Washington's radar. Gun violence came into sharp focus after the Sandy Hook Elementary School shootings that took the lives of 20 children and six adults.

It sparked a heated debate and led to new measures the president plans to put in place through executive action or with congressional approval.

Steps quickly assembled aimed at curbing gun violence, such as universal background checks and a ban on high-capacity magazine clips of more than 10 rounds.

"It's common sense" said Los Angeles Mayor Antonio Villaraigosa, who has been pushing for stricter gun laws. 'It makes sense. It's not a radical notion."

But the National Rifle Association and other gun groups are ready for a fight, swinging with searing TV ads and a public rebuke of the administration's efforts.

While polls indicate the majority of American's support the president's efforts, some political observers have doubts about Washington's ability to produce any meaningful change.

"No one Democrat or Republican wants to deal with that issue. It's not popular," said CNN contributor and conservative activist Erick Erickson. "They want to just do something for the sake of symbolism."

Debt ceiling

The new term will also feature some old fights. The debt ceiling round two was shaping up to be another fierce fiscal showdown. House Republicans had insisted on spending cuts in exchange for raising the debt ceiling. President Obama warned he was not going to negotiate.

With the prospect of a default and the Treasury Department taking steps to bridge the gap, Congressional Republicans backed off a bit, with plans to vote next week on a three-month deal to extend the debt ceiling.

It's not a long-term solution, but it takes some air out of a contentious battle the president would have faced immediately.

"I understand the games the Republicans are playing, but the game the American people want them to play is play fair," said Democratic strategist and CNN contributor Donna Brazile, who argued the rich should pay their fair share and struggling Americans should not be "disproportionately hurt" by deep cuts.

Immigration reform

One large and important constituency will be watching the president's second term closely. They care about gun violence and fiscal matters but want to see quick action on immigration reform.

Hispanics, who voted overwhelming for Obama, 71%-27% over Republican challenger Mitt Romney, had hoped for more progress in the president's first term.