Secretary of State Cyrus R. Vance today begins two weeks of travel that will include launching the next phase of Egyptian-Israeli peace negotiations and attempting to prevent a split in U.S.-British cooperation on dealing with Rhodesia.

Vance's first stop will be London, where his top priority will be to discuss with the foreign secretary, Lord Carrington, the chances for preserving the joint Anglo-America plan that seeks to include all of Rhodesia's bitterly feuding factions in a solution to the bloodshed there.

The plan's status has been clouded by increasing signs that Britain's new Conservative government may pull out of the joint approach and recognize the recently elected biracial government in the Southern Africa country.

Such a move would deal a major blow to the Carter administration's hopes of bringing the antigovernment guerrilla forces into a solution. It would increase the already enormous pressures buliding up in Congress for President Carter to lift sanctions against Rhodesia and recognize the new government to be headed by Bishop Abel Muzorewa. The Senate last week urged Carter to lift the sanctions.

From London, Vance will go to the Middle East for talks with Egyptian President Anwar Sadat and Israeli Prime Minister Menachem Begin. He then will represent the United States at the beginning of negotiations to create an internal self-government system for the Plestinian inhabitants of the Israeli-occupied West Bank and Gaza Strip.

These negotiations are expected to be even more difficult and time-consuming than the agonizing, top-level effort that was required to produce the Egyptian-Israeli peace treaty.

By being present at the opening of the new talks and at the turnover this week of Israeli occupied El Arish in the Sinai Peninsula to Egypt, Vance intends to underscore the administration's priority interest in the drive toward a comprehensive Middle East peace.

Vance also wants to make clear, particularly to the Arab world, that the appointment of presidential trouble shooter Robert S. Strauss as the new U.S. mediator for the Middle East talks does not mean any lessening of his or Carter's personal commitment to the success of the negotiations.

In another attempt to dramatize that commitment, Strauss, in a talk with reporters made public today, revealed that he plans to make his first Mideast trip in his mediatory role toward the end of June. Previously Strauss, who is winding up his assisgnment as Carter's trade negotiator, had said he didn't plan to become actively involved in the Middle East talks until autumn.

In fact, Strauss added, he would have accompanied Vance to this week's opening of the talks except for a prior commitment that will take him to China next week on trade matters.

Vance's itinerary also will include visits to Rome, where Italy's christian Democratic-led government soon must face an election challenge from the strong Communist Party, and to Madrid, where he will confer with the new government preparing to lead Spain into democracy.

In addition, Vance will attend the annual spring meeting in The Hague of North Atlantic Treaty Organization foreign ministers. His chief aim at the NATO session will be to brief West European allies intensively on the Strategic Arms Limitation Treaty with the Soviet Union and on the Vienna summit where the SALT agreement is to be signed by Carter and Soviet President Leonid I Brezhnev.

But the priority items on his agenda are the Middle East and Rhodesia.

In the Middle East, Washington is very anxious to demonstrate to those Arab contries maneuvering to isolate and penalize Sadat for his rapprochement with Israel that the U.S. mediated peace process offers the best hope of resolving the thorny Palestinian issue.

The new talks already have been clouded by Defense Minister Ezer Weizman's protest resignation last week from Israel's negotiation team. He reportedly was infuriated by statements attached to Begin's autonomy proposal that rejected the possibility of eventually creating a Palestinian state out of the West Bank and Gaza Strip and that declared Israel's intention ultimately to claim sovereignty over these areas.

However, Strauss, who talked with reporters Friday night, said it is expected that both sides "are obviously going to start out with some pretty extreme positions as a bargaining tactic" and added that it is far too early to predict how the negotiations will go.

He did concede, though, that there are some "negative trends" in the Middle East situation, such as greater-than-anticpated hostility being directed at Egypt by the other Arab countries and an upsurge of anti-Israeli terrorism. These, he said, were among the factors that persuaded him to hasten his involvement.

With regard to Rhodesia, the United States and the previous Labor government in Britain had been working to bring the government and its guerrilla foes together in a peace conference on a black majority government.The Carter administration's argument is that a lasting settlement is not possible without taking the guerrillas into account.

However, Britain's new prime minister, Margaret Thatcher, campaigned on a promise to recognize the Rhodesian government if