President Reagan proposed $13 billion in federal spending cuts and $3 billion in new revenues through elimination of tax abuses last night in what he called a difficult and painful but essential effort to stay on his promised course toward a balanced budget in 1984.

The president also abandoned his intention to lead the politically treacherous effort to restore Social Security to fiscal health, backing away from the benefit cuts he proposed earlier this year and throwing a challenge to Democrats who control the House of Representatives to join in a bipartisan search for a solution that would "remove Social Security once and for all from politics."

In a nationally televised speech from the Oval Office, Reagan said "the important thing now is to hold to a firm, steady course." While he said his new cuts keep alive the prospect of a balanced budget by 1984, his oft-stated goal, his address was strewn with indications that the political course to that goal is becoming more turbulent.

He proposed saving $8.4 billion next year by a 12 percent across-the-board cut in appropriations based on his March budget request. But since Congress exceeded his request for some programs, some of these new cuts would end up much larger than 12 percent, and would reopen bitter battles on Capitol Hill.

Reagan also proposed cuts in the more or less automatic benefit or entitlement programs in the budget not subject to the appropriations process. He said these would save $2.6 billion in 1982 and $15 billion by 1984, but that they will not be spelled out until later this fall.

Even these unspecified cuts, however, seemed to promise slices in programs Reagan told Congress in his first budget speech Feb. 18 would be immune--the programs he called "the social safety net."

In addition to the entitlement cuts and 12 percent slash in appropriations, Reagan proposed:

Reduction of the non-defense federal payroll by about 75,000 people over three years, in part by attrition, in part through firings.

The rapid dismantlement of the Departments of Energy and Education as well as the elimination of "a few smaller agencies and a number of boards and commissions, some of which have fallen into disuse or which are now being duplicated. Reagan will propose legislation by mid-October to abolish the Education Department and will complete a plan for dismantling the Energy Department by mid-November, a high administration official said.

* Reduction of about $20 billion in federal loan guarantees.

* Tax code revisions still under review to raise $3 billion that will affect some residential and business energy tax credits, tax-exempt industrial development bonds, co-insurance arrangements by life insurance companies, the unemployment compensation tax threshold and corporate tax collections.

* User fees for Coast Guard services, maintenance of rivers and ports, services to commercial airlines and private pilots which would recover $980 million. Reagan pointed out that he proposed these fees in March and Congress has not acted on them.

As he launched his new proposals into a political arena where Democrats are feeling newly restored strength and some liberal Republicans are objecting that they cannot support the president because he has cut only $2 billion from 1982 defense spending while slashing social programs far more sharply, Reagan warned that this is not the end.

There are still unidentified cuts of $11.7 billion in 1983 and $23 billion in 1984, according to his new budget figures, and worsening economic conditions or congressional failure to heed Reagan's guidelines could raise these totals.

"Let me be clear that this cannot be the last round of cuts," Reagan said. "Holding down spending must be a continuing battle for several years to come."

Reagan blamed Congress for creating the need for some of the new cuts, doubling back on the warm words the Reagan team directed at Congress in July after the budget and tax bills won Capitol Hill's approval.

He flinched from his assertion last week that cost-of-living increases in Social Security and other entitlements programs would be postponed one time next year. Republican congressional leaders had reacted sharply, warning the president that such a program would be defeated in Congress.

Reagan protected this latest round of budget cuts from some immediate political criticism by leaving some areas vague.

"I believe we've chosen a path that leads to an America at work, to fiscal sanity, to lower taxes and less inflation. I believe our plan for recovery is sound and it will work," Reagan said.

He appealed to the people to pressure Congress to back his program. Reagan asked them "to let your representatives know that you'll support them in making the hard decisions to further reduce the cost and size of government."

The president said he intends to set an example for the government by reducing the size of the White House staff. Word had been passed before Reagan spoke that people in the president's executive office would have to be fired.

There is no alternative to his plan, Reagan said.

He accused some of his opponents of having "participated over the years in the extravagance that has brought us inflation, unemployment, high interest rates and an intolerable debt." These well-intentioned but costly federal programs, Reagan said, "didn't eliminate poverty or raise welfare recipients from their dependence to self-sufficiency, independence and dignity. Yet in their objections to what we've proposed," Reagan said of his critics, "they offer only what we know has been tried before and failed."

The first signs of opposition came from a bipartisan group of governors who visited Reagan early in the afternoon.

Although they were given assurances that the threatened phaseout of revenue-sharing and curtailment of Medicaid and welfare programs had been dropped from the package, the governors said they would still fight to block the heart of the proposal, the 12 percent cutback.

Republican Richard A. Snelling of Vermont, the chairman of the National Governors Association, said that $7 billion of the projected $17 billion in savings over three years would come from programs of aid to the states. He said the governors told Reagan the states could not absorb further cuts this year without "ripping big holes in the safety net."

According to others in the meeting, Snelling told the president that forcing cuts of this scale three months into most states' fiscal year "could result in a public backfire" against the whole Reagan economic program.

Initial reaction on Capitol Hill was unenthusiastic. Senate Finance Committee Chairman Robert J. Dole (R-Kan.) said it would be "very difficult" to pass the package. He suggested that a 12 percent cut in domestic appropriations was too big and that more should be taken from defense. "It's got to be a balanced package," he said.

Senate Majority Whip Ted Stevens (R-Alaska) said Congress might have to cut $4 billion or $5 billion from defense, but he said Reagan could achieve his overall level of cuts.

House Minority Leader Robert H. Michel (R-Ill.) was more positive, saying, "I can make a good case for" the proposals. But he added, "I never saw a defense budget that couldn't be cut."

Democrats were predictably negative. House Majority Leader James C. Wright (D-Tex.) said, "You simply cannot have to largest peacetime military buildup in the history of our country and the largest tax cut and a balanced budget."

Senate Minority Whip Alan Cranston (D-Calif.) said, "The Congress will not give him the budget cuts, most of them unspecified, that he seeks."

Reagan introduced the portion of his speech on Social Security by claiming that "there has been a great deal of misinformation and for that matter pure demagoguery on the subject of Social Security."

Reagan, who proposed in May ending the $122 monthly minimum benefit and penalizing those who choose to retire early, defended those ideas while backing away from them. He said it was never his intention to take the minimum benefit from "those who truly need it."

Reagan said he now favors continuing the minimum benefit, although aides said that anyone with a private income exceeding $7,500 per year for a couple would not be eligible.

A high government official said these proposals would be dropped if Congress quickly gave the administration the power for borrowing between the various funds as a temporary measure to keep the system afloat and agreed to Reagan's appeal for a blue-ribbon, 15-member task force to review all the options and draw up a plan to insure the fiscal health of the Social Security system.

Reagan also defended his suggestion for a one-time postponement of the cost-of-living increase for Social Security recipients, although he abandoned the proposal. If this and his other changes had been made, he argued, "we would have solved the short and long-range problems of Social Security funding once and for all."

He blamed the Democratic leadership in the House for refusing to join in a "cooperative effort" on Social Security, saying that his feet were never embedded in concrete on the cost-of-living proposal.

Reagan made no rhetorical attack on Wall Street, which by keeping interest rates high and the markets down has expressed skepticism that the Reagan economic program will work.

The new cuts, Reagan officials have made clear, are in part aimed at persuading Wall Street that the program will work. Informed sources saids Office of Management and Budget Director David A. Stockman told one group yesterday that the cuts were needed to avoid the "credibility gap" the Carter Administration encountered in Wall Street as a result of its "unchecked spending machine."