WASHINGTON — President Biden said on Monday that his administration was on pace to achieve two key goals by March 25: 100 million shots of Covid-19 vaccines since his inauguration and 100 million direct payments under his economic relief bill.

The announcement was the first in what promises to be a series of end-zone dances that Mr. Biden and administration officials are set to stage this week as they promote the $1.9 trillion package that the president signed into law last week.

“Shots in arms and money in pockets. That’s important,” Mr. Biden said in a brief address from the White House. “The American Rescue Plan is already doing what it was designed to do: make a difference in people’s everyday lives.”

Over the weekend, the Treasury Department began issuing direct electronic payments of $1,400 per person, as authorized by the law, to low- and middle-income Americans. The United States has administered 92.6 million vaccine doses since Jan. 20, when Mr. Biden took office, according to data released on Monday by the Centers for Disease Control and Prevention. At the current pace of vaccinations, the country will pass 100 million doses before the end of the week, well ahead of the president’s promise of March 25.

Mr. Biden had set the goal of 100 million doses before taking office, and he has repeatedly heralded the country being on pace to meet it, though many public health experts say it is relatively easily attainable.

The relief plan also includes dozens of other provisions that have yet to be carried out, such as new monthly checks for parents, $350 billion for state and local governments and additional relief for the unemployed.

With so much money at stake and Republicans criticizing the package as wasteful, Mr. Biden vowed to bring “fastidious oversight” to the relief bill in order to ensure that it is distributed quickly and equitably.

He introduced Gene Sperling, a longtime Democratic policy aide who advised Mr. Biden’s presidential campaign last year, as his pick to oversee spending from the relief package. Mr. Sperling will be a senior adviser to the president and a White House employee, operating independently from an oversight commission established by Congress during the pandemic that consists of inspectors general from various agencies.

“We have to prove to the American people that their government can deliver for them, and do it without waste or fraud,” Mr. Biden said.

His remarks came as his team prepared to fan out across the country for a week of sales pitches for a bill that has proved very popular with voters but garnered zero Republican votes.

Mr. Biden will visit Delaware County, Pa., on Tuesday and appear with Vice President Kamala Harris on Friday in Atlanta, which helped deliver Democrats the Senate majority that made the relief plan possible.

A group of administration officials, including the first lady, Jill Biden, and Ms. Harris’s husband, Doug Emhoff, will make their own trips. Ms. Harris and her husband landed in Las Vegas for an event on Monday afternoon, while Dr. Biden finished an event in New Jersey.

The road show is an effort to avoid the messaging mistakes of President Barack Obama’s administration, which Democrats believe failed to continue vocally building support for his $780 billion stimulus act after it passed in 2009. The challenge for the Biden administration will be to highlight less obvious provisions, including the largest federal infusion in generations of aid to the poor, a substantial expansion of the child tax credit and increased subsidies for health insurance.

Mr. Sperling’s challenge will be to meet Mr. Biden’s promises of transparency and accountability for those programs.

The president and White House officials called Mr. Sperling well qualified for the task. He was the director of the National Economic Council under Mr. Obama and President Bill Clinton. In the Obama administration, where he first served as a counselor in the Treasury Department, Mr. Sperling helped to coordinate a bailout of Detroit automakers and other parts of the administration’s response to the 2008 financial crisis.

He advised Mr. Biden’s campaign informally in 2020, helping to hone the campaign’s “Build Back Better” policy agenda. Friends have described Mr. Sperling in recent months as eager to join the administration; he had been mentioned as a possible appointee to lead the Office of Management and Budget after Mr. Biden’s first nominee for that position, Neera Tanden, withdrew amid Senate opposition.

Mr. Sperling’s challenge with the rescue plan will be different than the one Mr. Biden faced in 2009, because the relief bill differs starkly from Mr. Obama’s signature stimulus plan. The Biden plan is more than twice as large as Mr. Obama’s. It includes money meant to hasten the end of the pandemic, including billions for vaccine deployment and coronavirus testing. The plans also have similarities, including more than $400 billion each in total spending for school districts and state and local governments.

Oversight of the $1.9 trillion relief legislation is currently expected to rely on the byzantine oversight architecture that was established in the stimulus packages Congress passed last year.

The new effort will continue to rely on the Government Accountability Office and the Pandemic Response Accountability Committee, a panel of inspectors general from across the federal government.

Less clear is the fate of the Congressional Oversight Commission, the five-person bipartisan panel that was created to oversee the $500 billion Treasury Department fund that supported the Federal Reserve’s emergency lending programs and loans to airlines and companies that are critical to national security. The commission currently has only three members, and the Fed programs concluded at the end of last year.

The commission’s report in January said that it planned to continue “analyzing loans, loan guarantees and investments that were made prior to program termination” and producing reports.

It is not clear if the existing mechanisms will be sufficient for overseeing the money in the new relief package, which will pump billions of dollars into states and cities. Additional oversight measures are likely to be needed.

A Treasury official said that the department would set up a process to monitor the use of funds that are being sent to states to ensure that they are used according to the eligibility requirements in the law.

Like many Americans in the pandemic, Mr. Sperling will have to coordinate and navigate those efforts virtually, at least at first. Jen Psaki, the White House press secretary, said on Monday that Mr. Sperling would work remotely from his home in California until he is vaccinated.