But on Sunday US Representative Heath Shuler, a Democratic moderate, mounted a challenge against Pelosi for the position, although he told CNN he would not likely beat her in the vote for House leadership, which is expected on Wednesday. No one else in the party has challenged her so far.

Shuler billed Pelosi's leadership as unacceptable after Democrats' big losses in the House in last week's elections and added that he would seek to take the party in a more centrist direction, as Pelosi has been criticized as being far to the left of many Americans.

Tax cuts and presidential leadership

The first item on Congress' plate will be whether to extend the tax cuts implemented by former President George W. Bush. Republicans want to extend the cuts across the board, whereas Democrats want to exclude the top 2 percent of earners.

The ultimate resolution may be a two year extension of all of the tax cuts, and the White House seemed to indicate that the President might agree to this, Fortier noted.

But whether that compromise will pass is going to ride on U.S. President Barack Obama' s leadership, Fortier said.

Pushback from his congressional Democratic allies has been strong on a two-year extension of across-the-board cuts.

So watching how Obama handles his own caucus and how and when he chooses to cut a deal with the GOP will be the first test of his leadership in a changed political landscape, Fortier said.

Ultimately, the president will probably have to adopt a mixed stance, finding some areas on which to compromise with Republicans or at least not to fight too directly with some other areas where he can draw contrasts with Republicans, Fortier said.

No easy ride for GOP

As a whole, polls show that while voters kicked out the Democrats, they are not fond of Republicans either.

With control of the House, Republicans have to do more than double down on a strategy of opposing the president and Democrats, McConaghy said.

"With statements that their top priority is to make President Obama a one-term president and other confrontational rhetoric, they risk misreading the national mood," he said.