President Reagan launched his long-promised effort to reinvigorate the American economy and the national spirit last night with an enthusiastically received appeal to Congress to trim the federal budget by $41.4 billion and lighten the burdens for individual and businss taxpayers.

In a nationally televised address to a joint session of Congress, Reagan voiced a grim forecast of the price of inaction, but an optimistic prediction that his far-reaching program will bring economic recovery.

"The people are watching and waiting. They don't demand miracles, but they do expect us to act.Let us act together," the president told Congress in a plea for rapid and favorable consideration of his plan. It was only the beginning of what a senior aide said would be "a full-court press" by the president on Congress not to rip his package apart.

White House officials said the president and his aides believe Reagan's plan has its best chance of winning congressional approval if it is considered quickly and as a whole rather than item by item. Senate Republicans were moving last night to handle the plan in keeping with Reagan's wishes.

Reagan's reception indicated the Congress is eager to work with him. He was welcomed with an extraordinary outburst of cheers and applause as he entered the House chamber, and his speech was interrupted 16 times, including a prolonged ovation when he asked Congress to join with him in making the program "our plan." Reagan stopped, smiled and said: "I should have arranged to quit right here." They applauded again.

The applause was much louder for his spending-cut proposals than for the tax cuts, however, and from Republicans rather than Democrats. Senate Minority Leader Robert C. Byrd (D-W.-Va.) said immediately after the speech that Democrats could not support the president's tax cut proposals. t

Taxation, Reagan said, "must not be used to regulate the economy or bring about social change. we've tried that and surely must be able to see it doesn't work."

Until its closing minutes, the speech was packed with details, in effect a no-frills summary of Reagan's proposals, which he calls "America's New Beginning: A Program for Economic Recovery."

The program slows the growth of future federal spending and beginning next year it puts dollars that would have gone for taxes back into the pockets of private and business taxpayers. The hope is to balance the budget by 1984, reduce inflation and simultaneously lower the unemployment rate. Reagan said the program would "create 13 million new jobs, nearly 3 million more than we would without these measures."

The plan cuts a long list of programs for the poor, and America's urban areas, but it also slices a number of business subsidies.

Only the Defense Department's budget rises. The ax falls on federal employes' benefits, eliminates the $3.5 billion Comprehensive Employment and Training Act's public jobs program, imposes user fees for commercial and recreational boat owners and proposes a vast reorganization and reduction in grants for education and social services, while also trimming food stamp spending.

The president attempted to build a defense against possible charges that the package of spending cuts penalizes the poor more than the rich by stressing that a $215 billion "social safety net," -- including the basic Social Security retirement benefits, Medicare and veteran's benefits -- would remain untouched by his reductions in government spending.

Reagan's plan was also designed to appear fair. "I hope I've made it plain that our approach has been evenhanded, that only the programs for the truly deserving needy remain untouched," Reagan said.

Despite his efforts, however, critics are certain to point to the welfare reductions and changes in education and other social service benefits as an unfair tilt that makes the non-elderly poor the largest losers from the Reagan package.

One radical change Reagan proposed potentially affects all wage earners. He called for a reduction to 13 weeks from the present 26 in which a worker can receive unemployment insurance while seeking a job comparable to the one he lost.

After 13 weeks, the government will consider any job suitable for the unemployed worker that pays the minimum wage or as much as his unemployment insurance. For a relatively well-paid auto worker who was laid off, the new system, which would take effect Oct. 1, 1982, would bring what the Reagan program drily described as "a change in work search practices" and a hoped-for more rapid reemployment. a

Only the Pentagon will enjoy a budget increase over the spending levels Reagan inherited from former president Carter. Military spending will go up $169.5 billion through 1986 to deal with what Reagan called "the threat we will face during the coming decade."

Since Pentagon spending will rise, as will the cost of the untouchable "social safety net," all of the spending cuts will come from the 29 percent of the 1981 budget marked for other programs. Reagan's plan projects this sector shrinking to 18 percent of the budget in 1984.

Dramatic as Reagan's proposals are, all the cost savings required for the future have not yet been identified by Office of Management and Budget Direct David A. Stockman and other planners. Further cuts of $80 billion for 1983 and $104 billion for 1984 will be needed. But areas to be cut $21 billion in 1983 and $30 billion in 1984 have not yet been found, according to the president's detailed presentation.

The reduction would, however, reverse the trend which has seen federal spending steadily become a larger part of the American economy. Reagan projects that after peaking at 23 percent of GNP this year, federal spending would decline until it becomes only 19.3 percent of the gross national product in 1984.

The budget cuts for 1982 total $41.4 billion. But the president's program would also raise $2.2 billion from users' fees affecting barge operators, airline passengers and boat owners plus an additional $5.5 billion in savings of off-budget outlays (as in many government lending programs), bringing the total reduction to $49.1 billion. This is almost exactly the $50 billion figure that Stockman called for and many critics immediately labeled unrealistic when the Reagan administration began its search for ways to pare spending.

The Reagan tax reductions of 10 percent a year for three years and greatly accelerated depreciation allowances for businesses will cost an estimated $53.9 billion in lost revenue for 1982, $100 billion for 1983 and $148 billion for the next year.

In the last stages of preparing the economic proposals, the president overruled some of his chief economic advisers and rejected a cut in the maximum tax rate on unearned income from 70 percent to 50 percent. A White House aide explained:

"A large number of his advisers presented the case to the president that the plan he first presented in the campaign was fair and equitable and to change now would persuade people that he was favoring the rich. If he made that change, his advisers said, there was a real chance that he would put his entire program in jeopardy."

Reagan opened his speech to Congress last night with a phrase he has used repeatedly since the Oct. 27, 1964, speech for Barry Goldwater that propelled him into elective politics. Again and again, the president has called his nation the "last, best hope of man on earth."

His speech and his detailed program draw a distinction between the American worker and those who are seen benefitting upjustly from government programs at the expense of their fellow citizens.

"The substance and prosperity of our nation is built by wages brought home from the factories and the mills, the farms and the ships. They are the services provided in 10,000 corners of America; the interest on the thrift of our people and the returns from their risk-taking. The production of America is the possession of those who build, serve, create and produce," Reagan said.

In addition to spending and tax reductions, Reagan's plan calls for elimination of thousands of federal regulations to achieve his campaign pledge "to get government off the backs of the people."

All three elements are interlocked and essential, the president said.

"It is time to create new jobs, build and rebuild industry, and give the American people room to do what they do best," Reagan said.

He told Congress that he wants other tax changes later, but deferred them because of the urgency he places on the program he delivered last night.Reagan aides have said they wanted "a clean tax bill" that would not tempt members of Congress to odd theri own pet tax projects, perhaps leading to a protracted Capital Hill debate.

As he did with his spending cuts, Reagan tried to anticipate criticism of his tax cuts. "Some will argue, I know, that reducing tax rates now will be inflationary," the president said. "A solid body of economic experts does not agree."

Reagan's speech, one of the most carefully worded and data-filled addresses of his political career, was written by the president, speechwriter Ken Khachigian and Stockman.Khachigian wrote a first draft, and then Reagan made substantial revisions with the help of Stockman and other advises.

Although the president plans to press Congress to act quickly on the program, he departs today for a four day vacation at his California ranch. The vacation trip, some sources argued, would weaken the message of urgency he delivered last night. However, Reagan and his closes aides felt the political risk was worth it to give the president the relaxation he wants.

Reagan does, however plan to telephone members of Congress during his vacation. He said repeatedly during the week that he is optimistic about the chances of his program on Capitol Hill.

In illustrating his contention that the economy is out of control, Reagan turned to a dramatic image.

"Our national debt is approaching $1 trillion," he said. A trillion, he added, is a number so large it is difficult to comprehend.

"I've been trying to think of a way to illustrate how big it really is," he said. "The best i could come up with is to say that a stack of $1,000 bills in your hand only four inches high would make you a millionaire. A trillion dollars would be a stack of $1,000 bills 67 miles high."

The president also left a challenge to anyone in Congress who would on pose this dramatic attempt to change Washington's normal way of governing by a president still buoyed by the landslide victory that carried him into office. To those who express unwill ingness to adopt his program, Reagan asked a question.

"Have they an alternative which offers a greater chance of balancing the budget, reducing and eliminating inflation, stimulating the creation of jobs, and reducing the tax burden?"