While the organizers of President Reagan's second inauguration were arranging the fireworks and parties that make the quadrennial event a combination of senior prom and county fair, other groups went public last week in an academic-intellectual version of an oriental bazaar, hawking ideas for Reagan's second term.

As if to dramatize how Reagan has stimulated the national debate by changing the political agenda, four organizations representing several think tanks and interest groups offered a wide range of recommendations last week for the next four years as well as critiques of the president's first term.

Some areas of agreement among the four reflect the changes and debates of the first four Reagan years -- the need for more cooperation between government and the private sector, the need for tax reform and simplification, overhaul of education and job retraining programs and the need to strengthen the family as the basic unit of society.

But each group had its priorities.

The organizations are:

* The Cato Institute, a public policy study group oriented to the free market and limited government and which defines its philosophy as 19th Century classic liberalism. It generally endorsed Reagan's concepts and accomplishments in a volume entitled "Beyond the Status Quo: Policy Proposals for America."

The study's editors expressed concern that most analysis ignores "fundamental problems and solutions" such as whether government programs should continue to exist and not just be the subject of a reflex debate over how much to increase their budgets.

* The National Urban League, a community-based, predominantly black service and advocacy organization for such programs as education and job training, which urged the administration to "stop fanning the flames of racial polarization."

The league's proposals, part of its annual report, paint "a grim picture of the state of Black America" but chose to stress the hopeful side of the picture in an effort to establish a dialogue between black leaders and the Reagan administration. It celebrated "a new spirit in America that refuses to accept the prospect of a nation divided between a prosperous majority and an impoverished minority."

"I call on him President Reagan to take a handful of small steps that could begin to heal the breach between his administration and black people," said John E. Jacob, league president.

* The Urban Institute, a policy research and educational organization that studies social and economic problems and government policies and programs; it offered a detailed critique of the changes in governmental institutions as a result of the "Reagan Revolution."

The institute's report, "The Reagan Presidency and the Governing of America," focuses primarily on the long-term political and institutional impact of the Reagan administration and concludes that for all its political successes, the administration "failed to forge a durable governing coalition that could extend its program . . . and in some cases weakened institutions and processes that are important for the long-term operation of the political system."

"More than any recent president, Ronald Reagan has challenged prevailing notions concerning the scope and structure of government," the editors, John L. Palmer and Isabell V. Sawhill, said. "It is crucial that a systematic assessment be made of how the administration has sought to change the government and wider political system and how successful it has been."

* A pan-ideological group calling itself "The Committee on the Next Agenda," a first time ever, ad hoc collection of scholars from The Brookings Institution, The Heritage Foundation, the American Enterprise Institute, the Hoover Institution on War, Revolution and Peace, the Hudson Institute and the Institute on the Research of Economics of Taxation.

The CNA was organized by Thomas D. Bell Jr., president of the Hudson Institute, to offer recommendations to help Reagan reduce "the social and economic burdens that have been imposed by the expansion of government power . . . promote the growth and efficiency of private institutions," and reassume "the mantle of world leadership."

In 1981, Bell became concerned over "the reactive nature" of politicians and administration officials who "don't have the time to take a long look at their policies."

"Like so many of these projects, this was the result of a conversation on an airplane, in this case with Marty Anderson," he said. "I saw that all of these groups were closer than most people realized." Anderson is a former Reagan campaign and White House aide now with the Hoover Institution.

To a great extent, the four reports are responses to the changes in the national political debate that Reagan has wrought and were timed to coincide with the beginning of his second term.

The Cato Institute describes its proposal as "a policy agenda for President Reagan's second term" whose ideas "fit the profile of what younger Americans seem to be looking for: economic growth and opportunity, a commitment to the economy of the future rather than that of the past, social tolerance, clean air and water, a compassionate and realistic approach to the less fortunate in society, a solution to the arms race and a peaceful foreign policy."

This starts with a reform of Social Security -- "a monument to the welfare-state concept: teetering at the edge of bankruptcy, inflexible and increasingly a bad deal" -- that would allow young workers to opt out of the system by guaranteeing benefits to current retirees but creating a "new Super IRA program."

Workers could put part of their Social Security taxes into these IRAs and eventually put all of them into private plans, which would "protect the elderly from a future tax revolt, prevent staggering tax burdens on younger workers, inject hundreds of billions of dollars into the private economy and avert the economic and social catastrophe of a Social Security collapse."

Cato calls for a constitutional amendment requiring a balanced budget on the grounds that this is the only way to penalize members of Congress politically for voting for increased government spending instead of rewarding them, as the current system does. It doesn't go into details of how to cut $200 billion, the anticipated size of the next deficit, out of the federal budget.

"Beyond the Status Quo" celebrates the deregulation of the communications and information industries -- primarily the breakup of AT&T -- and urges "a move toward private property and free markets in communications as the best means to protect First Amendment rights . . . regulators who try to protect consumers by obstructing competition will in the end simply lock them into dependency on an inefficient monopoly whose rates will continually ratchet upward anyway."

The Committee on the Next Agenda generally agrees with Cato's emphasis on a market-based economic system. Other areas of agreement include tax reform, job training for "hard-core unemployed youth," strengthening family life and private alternatives to Social Security.

The CNA also urged strengthening the family -- "Divorce, separation, desertion, illegitimacy -- these are the four horsemen of dependency," it said.

The report also contended that "the expanded role and complexity of government" has produced a society that is "excessively litigious and a legal system that is now grossly overblown." It urged reforms that would allow paralegals and professionals such as bankers and realtors to replace lawyers in negotiating contracts, divorces, wills and real estate closings.

At the other end of the political spectrum, the National Urban League urged Reagan to support the civil rights bill, end the policy of "constructive engagement" that gives South Africa's "apartheid regime the security of America's friendship," support his own initiatives to help the poor such as tax breaks for businesses that establish in high-minority "enterprise zones" and elimination of taxes on below-poverty incomes and to "end the shameful exclusion of blacks from key policy positions."

The Urban Institute report concluded that while Reagan changed the terms of the political debate and increased "public confidence in the presidency and the political system," he missed some long-range opportunities.

"After its early victories, the administration failed to forge a durable governing coalition that could extend its program and often failed to recognize opportunities to build on existing support," it said. As a result, "it encountered stalemate on the budget and made little headway on its proposals for regulatory reform, private-sector initiatives and federalism reform."

After Reagan's initial legislative triumphs in 1981 and 1982, the report concludes, he failed to defeat "the system that allowed the people in 1982 to reinforce the partisan deadlock after only two years by strengthening the opposition party in Congress in the midterm election."