Jubilant Republicans last night asserted that President Reagan's landslide reelection victory gives him an overwhelming mandate to continue his programs. And they were encouraged that they will achieve their hope of becoming the majority party, since he and other Republicans ran so strongly with white middle-class and young voters who approve of his management of the economy.

They also hoped that Reagan's victory would carry in enough new Republican House members to give the president a working majority there, to add to the Republican-controlled Senate.

"I think these returns are a strong mandate for the president's, and the party's, policies," Rep. Jack Kemp (R-N.Y.) said. "The people have indicated very strongly what they want."

Sen. Gary Hart (D-Colo.), who gave Walter F. Mondale a tough fight for the Democratic presidential nomination this year, strongly disagreed.

"There's no mandate at all, because of the kind of campaign the president ran," he countered. "It was a personality contest. His handlers kept him away from the people, and he can't go to Congress and claim a mandate, because he never talked about concrete programs."

Both parties now face a period of internal conflict as they struggle for superiority in the wake of Reagan's reelection.

The Republicans' task is to consolidate gains from the election and minimize factional disputes as they seek a full-scale political realignment that would end their half-century as the minority party.

The outcome of the Senate and House races yesterday, as well as the presidential election, will have an obvious influence on the nature of these disputes and the Republicans' hoped for realignment.

The return of such conservative leaders as Sen. Jesse Helms (R-N.C.) and the strengthening of the Republicans in the House will affect the tone and direction of the political dialogue of the next two years.

The Democrats face a more profound question -- the future direction of the party, as the once-powerful New Deal coalition of labor, liberals, farmers, minorities and southerners has fallen apart as an increasing number of them have defected to the GOP in presidential elections.

As big as the stakes are, however, for the first time in memory, neither party has a clear favorite for its next presidential nomination.

The Republicans will be looking for a successor to Reagan who can hold the party's factions together the way he did.

"The secret to the Republican nomination is to control the right, but not be controlled by them, the way Reagan did," one GOP professional said last week.

One of the major fights looming for the Republicans is between the exponents of supply-side economics such as Kemp, and more traditional party leaders, such as Sen. Robert J. Dole of Kansas, when they come to grips with the enormous federal budget deficits next year.

The supply siders oppose any tax increase and advocate further tax cuts to stimulate economic growth, which they contend will generate enough additional revenues to eventually bring the budget into balance.

The traditionalists scoff that this is a lotus eater's pipe dream, and contend that the only way is to cut federal spending and raise taxes.

There will also be internal struggles between New Right factions which want the party to move on social issues, such as constitutional amendments prohibiting abortion and allowing prayer in public schools, and moderates who want to concentrate on economics and broadening the party's base.

This will be an important element in the party's courtship of young voters, with whom Reagan runs phenomenally well and who are obviously a key to the Republicans' hopes of a permanent voter realignment.

Voters under age 45, who now are a majority, are more conservative economically than their elders and respond to Reagan's economic recovery and emphasis on the private sector and individual opportunity.

They are more liberal culturally in matters of life style, however, and are more likely to be put off by the abortion and prayer issues pushed by the Moral Majority.

The Democratic Party is divided roughly between traditional New Deal-Great Society liberals such as Walter F. Mondale, Sen. Edward M. Kennedy of Massachusetts and House Speaker Thomas P. (Tip) O'Neill Jr. of Massachusetts, and post-New Deal "neo-liberals."

"We've got to get away from the idea that the New Deal programs are sacred, and remember that Franklin Roosevelt was a grand experimenter," said Bob Squier, a Democratic campaign consultant.

"Reagan has given us a perfect opportunity," he said. "He's chopped up a lot of what we had on the books and we can start over by asking what government should do, what it can afford to do and what the middle class wants us to do."

"We've got to start listening to them and get away from saying, 'Trust us, we'll raise your taxes' because we know best."

But the traditional liberals worry about this.

"I see a vacuum of leadership after O'Neill retires and the House and Senate leadership moves South and West," one northern Democratic senator said last week.

"We can't get a civil rights bill or controls on Central America through the Senate now and we depend on the House," he added.

Leading the 1988 Republican presidential list is Vice President Bush, who has the advantage of his position; but, in the opinion of some party professionals, he hurt himself with his erratic campaigning this year.

He is joined among the front-runners by two unsuccessful candidates from 1980, outgoing Senate Majority Leader Howard H. Baker Jr. of Tennessee and Dole.

Kemp and Lewis Lehrman, who narrowly lost his New York gubernatorial race to Gov. Mario Cuomo two years ago, also are expected to run, as is Gov. Pierre S. (Pete) du Pont IV of Delaware.

The Democrats will be looking for a candidate to lead them back to the dominance they enjoyed for so many years.

Hart and Jesse L. Jackson are the only candidates from this year who are expected to run again in 1988. Some party professionals are convinced that Kennedy is getting ready for another run, and Cuomo is also a strong possibility.

After them come a number of relatively unknown, but potentially bright, stars: Sen. Joe R. Biden Jr. of Delaware appears sure to run; and Sens. Bill Bradley of New Jersey and Sam Nunn of Georgia, and Govs. Bob Graham of Florida, Charles S. Robb of Virginia and Mark White of Texas are possibilities.

Some Democrats think Reagan's success poses a lesson their party can learn. Republicans like Kemp tell their audiences that the Democrats "used to be the party of hope and opportunity, but now they're pessimistic, inward-turning and protectionist."

To some Democrats, the problem is symbolized by the dominant role of organized labor in the nominating process.

"The Democrats have a problem with special interests and their rhetoric and pessimism," one party professional said this week.

"Groups, like women and blacks, have a stake in pessimism because it's a tactic to achieve their aims. But we've gone from being the party of the working man to the party of organized labor.

"For all of Reagan's shortcomings, he has touched some important chords. The Democrats take it badly that he quotes Roosevelt and Truman and Kennedy, but thematically he is optimistic and idealistic and sounds like FDR in appealing to the same dreams and drives."

Ann Lewis, political director of the Democratic National Committee, foresees at least three major initiatives by the national party.

"We'll be more explicit and detailed about individual opportunity and economic growth in our rhetoric, as well as in practice," she said.

"We'll get governors and other state and local officeholders to play more of a national role," Lewis said. "And we'll make more use of the national committee. Because of its institutionalization and the programs it's developed, it's more valuable than ever before."

Charles T. Manatt, the national chairman whom Mondale tried to unseat just before the party's national convention last summer, isn't expected to seek another term.

Speculation on his successor centers on Rep. Tony Coelho of California, chairman of the Democratic Congressional Campaign Committee, who might be made a general chairman of the party as Sen. Paul Laxalt of Nevada is for the Republicans.

Others possible candidates are Lynn Cutler, vice chairman of the DNC; Paul G. Kirk, a former aide to Kennedy; Nancy Pelosi, former chairwoman of the California Democratic party; Sharon Pratt Dixon, a member of the DNC executive committee from the District of Columbia; and Duane B. Garrett, a San Francisco attorney and fund-raiser for Mondale.

There are also reports that some Democratic governors want to get behind a candidate, and party sources say that outgoing Gov. Scott Matheson of Utah is another possibility.