The people who keep up President Reagan's living quarters at the White House will have to find a way to save $176,000 of their $4 million budget starting March 1. Vice President Bush will have to trim $5,000 out of the budget of his official residence at the Naval Observatory, and his Senate expense allowance will drop by $2,000.

Congress will have $2.5 million less to spend on mail for the rest of the year. The Supreme Court will have to come up with $543,000 in savings, the National Security Council $156,000 and the National Institutes of Health $112 million.

From the massive Pentagon budget to the modest Capitol Guide Service, a new era arrives this week as the first wave of automatic, across-the-board budget cuts under the Gramm-Rudman-Hollings law are spelled out. The goal is to cut $11.7 billion from this year's deficit, but the exercise may foreshadow much bigger spending cuts later this year if Congress and Reagan deadlock on how to reduce the federal deficit.

The first round of automatic cuts is expected to go into effect March 1. According to a compilation prepared Friday by the Office of Management and Budget and the Congressional Budget Office, domestic agencies will have their budgets cut about 4.3 percent and defense about 4.9 percent.

These cuts, however, must be swallowed in the final seven months of the fiscal year -- meaning that in many cases they will be larger, perhaps 6 percent or 7 percent. Because Congress exempted many programs such as Social Security from the automatic budget knife, the cuts fall on the one-fourth of total government spending that is not protected. Moreover, the $11.7 billion must be saved from outlays, or actual spending; a larger cut will come out of budget authority, which represents policy commitments and permission by Congress to spend the money for them.

In some areas, "the cuts get pretty large," a White House official said. If the process is triggered for next fiscal year, he added, "it will be many times more intense."

White House officials are closely watching the political reaction to this first wave of spending cuts for clues about whether it will create pressure for a budget deal with Congress later in the year or lead to a stalemate triggering a big second round of cuts before the mid-term elections this fall in which Republican control of the Senate is at stake.

On Wednesday, the two budget agencies officially report on their projections of the deficit and how the $11.7 billion in automatic cuts are to be spread throughout the government. When the two agencies disagree, they take an average of their estimates. Later, the president is expected to issue an order putting the cuts into effect March 1.

According to budget office documents, the cuts will start in the Oval Office. The White House Office will be cut by $951,000 of a $22 million budget.

Exactly how each cut will be made is to be determined by officials in each agency.

The Pentagon, in particular, was given special flexibility this year to protect some programs at the expense of others, but still must produce the overall savings.

Sources said one of the top "protected" defense items is Reagan's program to develop a space-based missile defense shield, the Strategic Defense Initiative, which was cut back by Congress this year.

Some other cuts outlined in the budget documents:

Congress, including House, Senate, Architect of the Capitol, Congressional Budget Office and other functions, $62 million. The judiciary, including Supreme Court, $37.7 million. Executive Office of the President, $4 million. Funds appropriated to the president, $268.7 million. Agriculture Department, $1.2 million. Commerce Department, $68 million.Defense Department (military) $5.1 billion, (civilian) $497 million. Education Department, $170 million. Energy, $333 million. Health and Human Services, $1 billion. Housing and Urban Development, $32 million. Interior, $201.8 million. Justice, $143 million. Labor, $171.5 million. State, $99 million. Transportation, $373 million. Treasury, $373 million.