Vice President Mondale leaves Washington today on a round-the-world trip, carrying a pledge to America's major allies that the Carter administration wants a "close and cooperative" relationship on the economic and political issues facing them in the next four years.

President Carter, emphasizing what his press secretary called "the particular importance" he places on the trip, has asked Mondale to a White House breakfast and a formal send-off before the Vice President takes off for Europe from Andrews Air Force Base.

Mondale is scheduled to visit Brussels, Bonn, Berlin, Rome, London, Paris and Tokyo before returning to Washington on Feb. 1 from his 10-day tour. Officials said they could not recall a similar journey by a President or Vice President so early in a new administration.

White House press secretary Jody Powell said yesterday that Carter had asked him "to reiterate the particular importance" he places on the trip and to stress to the foreign governments that Mondale is not just "a personal friend" but a "chief policy adviser."

Powell said the President views the trip as "symbolic of the close and cooperative relationship" he pledged to seek with America's allies, and as an "importance fact-finding and consultative mission."

The itinerary released yesterday shows Mondale will have a private audience with Pope Paul VI, meet with leaders of NATO and the European Economic Community, as well as principal officials of Belgium, the Netherlands, west Germany, Italy, Great Britain, France and Japan.

Economic issues are expected to be in the forefront, as the Vice President briefs allied governments on the new administration's economic stimulus" package and probes their own efforts to cope eith the twin problems of inflation and unemployment buffeting most of the industrial nations.

Powell said Carter stressed that "every family in the world" will be affected by these policies and "the need for cooperation . . . has never been greater."

In this sense, the Mondale trip is the first step in the process that will lead to what is now planned as Carter's own first international conferrence, an "economic summit" of industrial nations later this year.

Beyond the economic problems, the Vice President is expected to initiate discussions on a wide range of such sensitive subjects as West Germany's and France's plans to sell nuclear fuel reprocessing plants to Pakistan and Brazil; France's recent release of Palestinian terrorist Abu Daoud, and Japan's reaction to Carter's campaign pledge to withdraw U. S. troops and nuclear weapons from South Korea.

Carter aides stressed that the administration will have been in office less than 90 hours when Mondale opens his talks Monday in Brussels. They said the Vice President is "carrying to new proposals."

"We are not going to impose on our friends a grand design," one official said, "We are dead-serious about the consultation process."

Some of the stops are clearly designed to indicate the continuity of American policy - notably the visit to the Brandenburg Gate between East and West Berlin, where officials said Mondale will pledge the new administration to protect, "the basic rights and freedoms" of the enclave behind the Iron Curtain.

But they stressed that much of the discussion will be aimed at exploring allied governments' reactions to "changing the historical circumstances" and at communicating the Carter administration's readliness to re-evaluate past American policies in the light of altered conditions.

Issues on Mondale's agenda include: Soviet American relations, including both the strategic arms limitation talks and the discussions on mututal troop reductions in Europe.

North-South relations including the position of the northern industrialized countries on the issues of commodity prices and the transfer of technologies to less developed countries when the adjourned Paris talks resume.

Energy issues, including at least a general discussion of the Carter approach to the question and the possibilities of cooperative action among the energy-importing countries.

Trade and monetary policy questions, looking ahead to the new round of negotiations scheduled for this year.

Problems of specific including Cyprus, the Middle East and southern Africa.

Officials said that, in preparation for the trip, Mondale has met three times with Secretary of State Cyrus Vance and at least once with the secretaries of Defense, Treasury and Commerce and the chairman of the Council of Economic Advisers.

Mondale is traveling with a staff of almost two dozen foreign policy, economic and military advisers and personal aides. About 30 reporters are accompanying him on this first major diplomatic mission of the Carter administration.

Officials said the reason no stop is planned in Canada, a key country in the forthcoming economic talks, is that Prime Minister Pierre Elliott Trudeau is ecpected to visit Carter in Washington in the near future.