Tuesday’s big electoral wins have handed the reins of Congress to Republicans for the remainder of President Obama’s term, but immediately raise the question of whether the next two years will bring more acrimonious gridlock – or some compromise on key issues ranging from immigration to energy.

In his victory speech Tuesday night, Senate Republican Leader Mitch McConnell, who is expected to ascend to majority leader, said some things won't change next year. But he stressed that lawmakers and the White House don't have to be in "perpetual conflict" and "have an obligation to work together."

McConnell said Wednesday marks the start of “the race to turn this country around.”

Different flanks of the Republican Party have different ideas of how that effort would play out.

Some plan to use the perks of majority-hood to put renewed pressure on the Obama administration. With Republicans gaining at least 7 seats in the Senate, the party will chair key committees and have the power to launch investigations and hold hearings from both chambers. Plus they’ll be able to more easily hold up political and judicial appointments.

At the same time, the Republicans clearly will be short of a 60-seat, filibuster-proof majority, meaning they’ll have to draw at least some Democratic support to pass favored legislation. And they’d need to either work with Obama, or corral an even bigger coalition to override presidential vetoes. Some warn that passing bills just to make a point would merely give new meaning to the “party of no” and say they should concentrate on working with the president and Democratic leaders – and vice-versa -- to get something done in the next two years. Republicans so far have reached a 52-seat majority, with three races yet to be decided.

Analysts urged the party to start early on charting their agenda.

“This is the point where they need to show leadership ahead of the politics,” said columnist and Fox News contributor Charles Krauthammer. “Republicans should have an agenda where they pass a bill a week – start small, start now.”

One issue to watch: whether conservatives will push a repeal of the Affordable Care Act, which conservative Sen. Ted Cruz, R-Texas, told Fox News Tuesday night was at the top of his own list – but he also left open the door to smaller-scale efforts.

“Now that we have won the election it is incumbent on Republicans to stand up and lead,” Cruz said. “It is incumbent on us to deliver with bold, positive, optimistic agenda, focusing on jobs … economic growth and opportunity and focused on defending our constitutional rights.”

Repealing ObamaCare is a priority, he said, but if that doesn’t work he said lawmakers should go after smaller provisions of the bill. He also called for “meaningful oversight of the administration in the Senate,” with “serious, sober, systematic hearings” on issues ranging from Benghazi to the IRS.

The Republicans pulled off major victories in key states Tuesday, gaining 7 new Senate seats so far; they claimed the majority in the Senate and built their majority in the House. Louisiana's Senate race is heading to a run-off, but polls have shown Democratic Sen. Mary Landrieu in trouble in a head-to-head race against Republican Rep. Bill Cassidy. In Virginia, Democratic Sen. Mark Warner was clinging to a narrow, 1 percentage point lead over Republican Ed Gillespie with most precincts reporting. Ballots are still being counted in Alaska, but Republican Dan Sullivan is holding a few point lead over Democratic Sen. Mark Begich.

Aside from health care, also on the radar is whether Republicans will try to scale back any executive action taken by Obama to prevent millions of illegal immigrants from being deported.

Obama appears to be eager to get some sense on how things will go, already inviting top congressional leaders from both parties to meet with him Friday at the White House. For his part, outgoing Democratic Majority Leader Harry Reid said he is willing to work across the aisle. “I look forward to working with Senator McConnell to get things done for the middle class," he said in a statement Tuesday night.

On the issue of immigration, Republicans might have enough votes to use the appropriations process to defund any executive measures the president invokes to prevent illegal immigrants now in the country from being deported. They may also have the numbers now to push for tougher border enforcement legislation. But comprehensive immigration reform, which passed the Democrat-controlled Senate but failed in the House, may be less likely.

Former Republican presidential nominee Mitt Romney, speaking with Fox News Tuesday night, believes there are areas of immigration reform – like better security on the border – that Republicans can probably pursue piecemeal, if they are willing. “These issues will be raised … perhaps they will come to the president’s desk in bills, one by one, and hopefully the president will sign them.”

But there are other areas where Republicans may work with the president directly to get favorable legislation through his last two years of office. And there are plenty of areas of overlapping interests -- at the top, Obama’s desire to get special negotiating authority for a new Asia-Pacific trade pact. Up until now, Reid has blocked that authority. But with a new Senate majority leader, that could change.

Sensing that Obama might only need a little push – and maybe a few sweeteners -- Republicans could see their way to getting a long sought-after green light for the contentious Keystone Pipeline. The administration has delayed a decision for six years. Republicans may find an avenue for approval, perhaps in some larger energy bill that offers Obama and key Democrats a little grist, like increased coal emission standards.

According to Republican National Committee Chairman Reince Preibus, Republican senators will get it through with or without Obama’s help. “We will pass a budget in both chambers, number one, and we will pass the Keystone pipeline, number two,” Priebus reportedly declared Tuesday morning, before the Republican victories were even a reality.

“And I actually think the president will sign the bill on the Keystone pipeline because I think the pressure — he’s going to be boxed in on that, and I think it’s going to happen,” he said.

This recalls a similar dynamic when Democratic President Bill Clinton was in the White House and Republicans controlled both the House and Senate. Clinton had no choice but to work with the Republicans to get his – and their – pet legislative projections signed into law, including welfare reform and a balanced budget.

“I think if you see good policy going through the House and Senate and to Obama’s desk it will finally force him to do his job and make good decisions,” offered Rep. Steve Scalise, R-La., in an interview with Fox News.

“If they can get agreement between the House and Senate, they can force [Obama’s] hand on a whole host of things, making his life difficult,” former Sen. Evan Bayh, D-Ind., told Fox News.

But that would take into account that all Republicans want to work with the president. With the 2016 presidential race just two years away and a number of GOP senators putting out obvious signals that they are going to run for the nomination, chances are, much of the activity in the Senate these last two years will be seen through the lens of the campaign.

Rather than find areas of agreement in order to get their measures signed by the president, some of the chamber’s most conservative Republicans might seek to introduce bills they know the president will surely veto in order to draw bright lines and rally the base.

ObamaCare’s repeal will certainly be championed by the Republican majority in the House. Majority Leader Kevin McCarthy, R-Calif., said as much in an interview Tuesday night. “I think we would repeal ObamaCare and replace it.”

Pennsylvania pollster Terry Madonna warns not to expect too much. “You can’t find an analyst anywhere saying that the gridlock will change much. There is not likely to enough cohesion or bipartisanship to do anything,” he said. “I think it’s going to be a treacherous place to work.”

McConnell said this is the kind of thinking that must be changed. “Just because we have a two party system doesn’t mean we have to be in perpetual conflict,” he said.