Secretary of State-designate George P. Shultz began to take up the reins of U.S. diplomacy yesterday in a lengthy meeting at the State Department with the departing Alexander M. Haig Jr.

Shultz' initiation into the corridors of diplomatic power came amid redoubled U.S. concern about Israeli military action against Palestinian guerrilla forces in surrounded Beirut.

White House officials, including national security adviser William P. Clark, were taking part in an unusual Sunday session on the Lebanese situation even as Shultz and Haig were meeting at the State Department. The same subject is expected to dominate a meeting of the National Security Council, which has been scheduled by President Reagan for this morning at the White House.

Israel's statements yesterday, demanding that Palestine Liberation Organization forces get out of Beirut and advising all civilians to leave, appeared to set the stage for invasion of the capital or some other greatly intensified military action. One strand of U.S. policy in recent days has been an effort to avert such a bloody showdown in Beirut.

The developments in the Middle East caught the Reagan administration in a particularly difficult moment, in the first stage of the transfer of authority at the State Department from Haig to Shultz. The result was an awkward silence amid continuing uncertainty about the administration's Middle East policies, which figured in Haig's resignation.

The State Department had only the sketchiest reaction to Middle East developments yesterday, saying through a press spokesman that "we continue to hope that all parties will observe the cease-fire." The department evidently had not given up on the possibility that members of the PLO will agree through negotiation to lay down their arms, because reporters were told that the one-sentence statement was not intended to encourage the Palestinians to hold out.

A White House official, reacting to reports that mixed signals from Washington on the Middle East infuriated Haig and may have complicated negotiations with the PLO, denied that a separate White House channel had been established in recent weeks.

According to the official, a series of verbal and written messages from the White House to special U.S. Mideast envoy Philip C. Habib was undertaken with the full knowledge and participation of Haig and the State Department. Sources close to Haig, on the other hand, said Saturday that Haig had been bypassed.

The White House official described two meetings between presidential assistant Clark and Saudi Ambassador to Washington Faisal Alhegelan as being of an almost perfunctory nature, lasting about 10 minutes each, and both reported to the State Department. The first meeting, on June 19, reportedly flowed from a call on Nancy Reagan the same day by the ambassador's wife as part of an Arab wives' delegation concerned about the bloodshed in Lebanon.

In the second meeting, on June 22, Alhegelan urged that the U.S. government communicate with the PLO in order to reach a settlement, according to the White House official. The ambassador was simply told, by this account, that the administration was already doing that indirectly, through the intermediary help of the Saudi government.

A garbled account of the second conversation, quoting Clark as saying that Israel would pull back six kilometers and thus loosen its noose around Beirut, was picked up by Habib from Saudi officials in the Middle East. It was quickly refuted from Washington, according to the White House official.

The official acknowledged, nonetheless, that Clark's meetings with Alhegelan had been among the grievances cited by Haig last Thursday morning when offering his resignation to Reagan.

The full scope of Haig's reasons for resigning remains officially unexplained. Haig has not been seen in public since he read his letter of resignation at a Friday afternoon press conference.

His chosen successor, Shultz, has made only brief appearances in which he has had nothing to say about the substance of U.S. policy. Shultz has refused to answer questions, saying these will be his "days of silence."

Neither Haig nor Shultz was seen by reporters as they arrived separately at the State Department for the meeting that began about 2 p.m. There was no appearance by either man for reporters or even cameras either during or after their three-hour meeting, and the State Department had nothing to say afterward.

Reagan, returning to the White House from a weekend at Camp David, refused to answer questions about the Lebanese situation or the change at the State Department.

Despite Reagan and Haig's announcements of the resignation Friday, Haig is still secretary of state and will continue until a decision is made that an orderly transition is well advanced, according to a senior State Department official. Outgoing cables from the State Department yesterday were signed by Haig as chief U.S. diplomat.

Shultz will not officially take over until after the Senate votes on his nomination, which is not likely before mid-July at the earliest.

Sen. John Glenn (D-Ohio), a member of the Foreign Relations Committee, which will consider the nomination starting the week of July 12, said yesterday on "Meet the Press" (NBC, WRC) that he wants Haig to appear at the confirmation hearings to explain the statement in his letter of resignation that U.S. foreign policy is shifting away from a course of "consistency, clarity and steadiness of purpose."

Glenn also said he would like to explore Shultz' relationship with the Bechtel Group Inc., the giant international construction and engineering company that Shultz has headed since ending his Washington service in 1974 as secretary of the treasury in the Nixon administration. The Ohio senator said, however, that he anticipates that Shultz will pass the committee's tests.

Committee Chairman Charles H. Percy (R-Ill.) said on "This Week with David Brinkley" (ABC, WJLA) that Haig had told him in a telephone conversation he resigned because of "policy differences." These covered a wide range, including the controversy over the Soviet gas pipeline, the Middle East and the Falkland Islands, Percy said.