Senate Majority Leader Mitch McConnell spoke on Capitol Hill in Washington in December as House Speaker Paul Ryan and others listened. AP
WASHINGTON—The capital’s all-Republican leadership takes power this month with many common goals—but with different time frames and political constraints that suggest they may not always be in sync.

The morning after the GOP won the House, Senate and White House on Election Day, Senate Majority Leader Mitch McConnell cautioned Republicans against reading their sweep as license to push through a strictly partisan agenda.

“Overreaching after an election is, generally speaking, a mistake,” Mr. McConnell told reporters. He emphasized that Democratic votes would be needed to pass most legislation in the Senate. “I don’t think we should act as if we’re going to be in the majority forever,” he said.

On the other side of the Capitol, House Republicans, after years of chafing under a Democratic president, are pushing to make sweeping changes to the Affordable Care Act and other policies—and to make them faster than their Senate colleagues. But while Donald Trump’s election likely will enable much of the GOP wish list to become law, the unorthodox president-elect has shown he isn’t bound to traditional Republican ideas. That creates a potential for tensions between Mr. Trump, who has called both for tax cuts and a big package of infrastructure spending, and congressional budget hawks wary of deficit spending.

“The thing that we have to keep an eye on is the budget,” said Rep. Thomas Massie (R., Ky.). “We don’t want to have control of all three levers—the House, the Senate and the White House—and then actually increase the deficit.”

During his campaign, Mr. Trump called for $1 trillion worth of new infrastructure construction, generally relying on private financing, which industry experts say is likely to fall far short of funding it.

Congressional Republicans often agree broadly on issues such as repealing and replacing the Affordable Care Act, but quickly splinter over the details. For instance, many House Republicans are eager to fully overhaul the health-care law within the next two-year session of Congress, a timeline that many Senate Republicans believe is unrealistic.

The Senate is likely to frustrate House Republicans, as well as Mr. Trump, on other issues. Most big policy tasks, such as reworking the immigration system, passing spending bills and increasing the federal borrowing limit, will need 60 votes in the Senate. Because Republicans will hold only 52 of the chamber’s 100 seats, most bills will need bipartisan support.

House Republicans say they don’t plan to cede policy to the Senate. “We’re going to be cooperative but not submissive in the House,” said Rep. Dennis Ross (R., Fla.).

One potential risk for congressional Republicans is that Senate gridlock drives Mr. Trump to try to bypass Congress and act on his own through executive actions. That would raise the hackles of the Freedom Caucus, a group of roughly 40 conservative House Republicans, said one of its members, retiring Rep. Matt Salmon (R., Ariz.).

“If Trump gets really bored with waiting for the Senate to get things done and starts to act like Obama and do a bunch of executive orders, if the Freedom Caucus folks feel they’re unconstitutional—even if they love the policy—I think they’ll end up not falling in line,” Mr. Salmon said.

One ally conservatives will likely have in the administration is Rep. Mick Mulvaney (R., S.C.), Mr. Trump’s pick to be budget director. Mr. Mulvaney is a fiscal conservative who has in the past tried to block budget gimmicks deployed by other Republicans hoping to boost military spending. That could put deficit and defense hawks on a collision course in next year’s budget.

It could also put Mr. Mulvaney at loggerheads with Mr. Trump, who has promised to slash taxes while also spending more on defense and other items. Mr. Trump has said the economic growth produced by his policies would help shore up the country’s fiscal outlook.

Republicans in both chambers also are treading cautiously to avoid Mr. Trump’s Twitter-directed ire when they disagree with him on economic and foreign policy.

“We’re in a ‘two scorpions in a bottle’ phase,” said Stewart Verdery, the founder of lobbying and public-affairs firm Monument Policy Group and a former Senate Republican aide. “They know they have to coexist, and if they attack each other, they’re both going to die.”

Signs of tension could escalate this month, when many of Mr. Trump’s cabinet nominees will have confirmation hearings and votes in the Senate. Several GOP senators known for hawkish foreign-policy stances have voiced concerns over the business ties with Russia developed by Rex Tillerson, Mr. Trump’s intended nominee for secretary of state, during his tenure as chief of Exxon Mobil Corp.

Republican senators are also divided over how to pursue investigations into alleged Russian hacking during the U.S. presidential election, which Mr. Trump has dismissed as a partisan concern. Sen. Cory Gardner (R., Colo.) plans to introduce legislation creating a new, select committee to probe the cyberattacks this year. But Mr. McConnell has said the inquiry should be handled by the Senate intelligence panel, which conducts much of its business in secret.

Democrats may have to navigate their own tensions. The party will want to accommodate the political needs of centrists such as Sens. Joe Manchin of West Virginia and Joe Donnelly of Indiana, who are expected to run for re-election in 2018 in states that Mr. Trump won by wide margins, as well as consider policy proposals from the party’s newly empowered liberal wing. Leading liberals in the Senate, Bernie Sanders of Vermont and Elizabeth Warren of Massachusetts, now hold Democratic leadership posts.

One potentially sticky situation for Republicans will be managing the government’s debt limit, which GOP lawmakers have often been loath to raise. As part of a sweeping budget and debt deal passed in the fall of 2015, Congress suspended the debt ceiling until mid-March 2017. The Treasury Department could take steps known as “extraordinary measures” to enable borrowing until at least mid-summer, according to estimates by the Bipartisan Policy Center.

Mr. Mulvaney has voted in the past against raising the debt ceiling, and Republicans have typically needed Democratic support to pass legislation increasing the limit.

Write to Kristina Peterson at