Within hours of solidifying their control of Congress, Senate Minority Leader Mitch McConnell and House Speaker John A. Boehner were quietly laying plans for a series of quick votes in January aimed at erasing their obstructionist image ahead of the 2016 elections.

First up: Action on long-stalled bills with bipartisan support, including measures to repeal an unpopular tax on medical devices and approve construction of the Keystone XL oil pipeline.

After that: Pass a budget through both chambers of Congress for the first time since 2009, followed by the full array of government-funding bills. “There will be no government shutdowns,” said McConnell, who will soon become majority leader.

Finally: Aim for the big score. Not repealing President Obama’s Affordable Care Act, though the conservative campaign to undermine the law will proceed in the background. Instead, Republicans dangled the prospect of fast-track trade agreements and sweeping tax reform as potential areas of agreement during Obama’s waning days in office.

Even before voters went to the polls Tuesday, McConnell (R-Ky.) and Boehner (R-Ohio) began mapping this coordinated strategy in hopes of unifying Republicans, picking up support from some Democratic lawmakers and putting pressure on Obama to compromise or risk drawing his own charges of intransigence.

The GOP gained control of the Senate Tuesday night, taking hold of the legislative agenda in that chamber. Here are three of the policies Republicans are likely to tackle as they take the reins in January 2015. (Julie Percha/The Washington Post)

According to three Republicans who requested anonymity to speak freely about the talks, Boehner and McConnell plan to roll their agenda out over the next few weeks in a series of joint appearances, joint memos and joint op-ed articles — including in the Wall Street Journal on Thursday — promoting their decision to work together.

“Our fortunes are tied,” said Sen. John Thune (R-S.D.), who chairs the Senate GOP conference. “We’ve got to hang together, and that approach starts at the top with Mitch and Boehner.”

Added Sen. Orrin G. Hatch (R-Utah): “We’re going to have a Republican Senate that wants to govern.”

At the center of these discussions is a desire to avoid the kind of intraparty drama that has roiled Washington since 2011, when Republicans took control of the House with a freshman class dominated by tea-party conservatives.

Elated by Tuesday’s election results, Republicans remain burdened by the shutdown of the federal government last year, which sullied the party’s brand. And they still face the prospect of managing an unruly right flank more interested in challenging Obama than in following the leadership’s playbook.

Pressure to follow through on the party’s most conservative priorities was already building Wednesday. Several tea-party leaders gathered at the National Press Club and reiterated their demand that repealing the Affordable Care Act remain the party’s priority. In a memo, Sen. Ted Cruz (R-Tex.) also emphasized the health-care law as a top target.

“Now is the time to go after and do everything humanly possible to repeal Obamacare,” Cruz said in a speech late Tuesday in Texas.

Even the National Review, a magazine popular with the conservative elite, warned McConnell to not forget the activists and grass-roots conservatives who lifted him to power.

“With all due respect to the senator and like-minded Republicans, this course of action makes no sense as a political strategy,” the editors wrote. Though tax reform and trade are good ideas, they wrote, “voters are not, in fact, waiting anxiously for any of this. Business lobbies are.”

Other Republicans said McConnell and Boehner face a political imperative to establish themselves as leaders who can run their chambers effectively.

“There’s a real incentive to put things on the president’s desk that are reasonable,” said Glenn Hubbard, the chief economist in the George W. Bush White House and a top adviser to Mitt Romney’s 2012 presidential campaign. “He may not sign them all. But the hope is he would sign some of them.”

In the House, where GOP leaders have been plagued by conservative defectors, Boehner stands a good chance of consolidating his authority given his gains at the polls, said Sen. Richard Burr (R-N.C.), a close friend of Boehner’s. Republicans will number at least 243 in January — a gain of 14 — the biggest House GOP majority since the Truman administration.

Boehner “can sell himself as someone who was able to win and who is able to work with the Republican Senate,” Burr said. “That’s going to hopefully create some more trust among people in the House and create a desire to govern.”

Managing the new Senate majority could also be a challenge. In addition to Cruz, McConnell faces the prospect of as many as 24 Republican senators running for reelection or retiring in 2016 — nearly half a conference that could number 54 members. Many, such as Patrick J. Toomey of Pennsylvania and Rob Portman of Ohio, will be running in purple states and could be reluctant to take tough votes.

On the other hand, the class of Senate Republicans elected Tuesday may make McConnell’s job easier. Confidants say McConnell so far has few concerns about the newcomers and believes most are inclined to work with him. Several were able to overcome tea-party insurgents in the primaries and therefore do not fear the wrath of activists as much as some Capitol Hill veterans.

The class includes Rep. Shelley Moore Capito (W. Va.), a longtime Boehner ally in the House. Mike Rounds (S.D.) is a former governor. And Iowa state senator Joni Ernst was backed by her state’s Republican establishment.

“You know, I think the gridlock is going to end,” Sen. Rand Paul (R-Ky.) said in an interview Wednesday. “Senator McConnell knows how to make the Senate work and I think he’ll make us a bigger party, a better party, by . . . crafting bipartisan legislation.”

Paul, who is widely expected to seek the Republican presidential nomination in 2016, warned that conservative firebrands in both chambers should not expect the new GOP Senate to pursue an aggressive agenda on every front.

“The House needs to realize that to pass anything, we’ll need 60 votes in the Senate, and getting there is the obstacle,” he said.

Indeed, the very Democrats most likely to vote with Republicans were the ones who fell to defeat Tuesday. With moderate Democrats such as Mark Pryor (Ark.), Mark Begich (Alaska) and Mark Udall (Colo.) gone, Republicans will find a much smaller pool of potential allies when they reach across the aisle.

To overcome that hurdle, Republicans plan to advance narrow legislation that already has some Democratic support. While pursuing a quixotic campaign to repeal the Affordable Care Act, for example, GOP leaders are discussing more limited efforts to undermine the law, such as watering down a requirement that employers offer full-time workers coverage, which takes effect in January. The administration has already delayed portions of the mandate.

They could also try to change the definition of a full-time worker from someone who works at least 30 hours a week to someone who works at least 40 hours a week, another proposal likely to draw Democratic support.

For more ambitious legislation, senior Republicans are discussing a fast-track procedure known as “reconciliation,” which would permit them to push a measure through the Senate with just 51 votes. Among the targets under consideration: tax reform, cuts to Medicare benefits and an increase in the federal debt limit.

But former senator Judd Gregg of New Hampshire, the last Republican to pass a reconciliation bill, said he is warning friends in the Senate that the procedure is politically dangerous unless the president is willing to sign the resulting measure into law.

“A lot of people out there seem to think it’s a magic bullet. It’s not,” Gregg said. “The magic bullet is a bipartisan bill. That’s how you do it without causing yourself huge political damage.”