Israeli Prme Minister Menachem Begin and Egyptian President Anwar Sadat opened talks at this seaside resort tonight in a display of harmony designed to speed normalization of Egyptian-Israeli relations and give new direction to the thorny negotiations on Palestinian autonomy.

Begin's visit here, where Sadat maintains a palatial summer residence, marks his fourth official stay in Egypt, after trips to Ismialia in 1977 and to Cairo and El Arish earlier this year.

It seems to have less drama than the ground-breaking earlier journeys, however, and is more of a business meeting between the leaders of two nations drawing steadily away from the hostility that marked their relations for a generation.

But the irrepressible Egyptian people nevertheless touched Begin's first day of a planned three-day visit with apparently spontaneous warmth, mobbing his car and pressing to shake his hand as he left an Alexandria synagogue where he had prayed and greeted the small congregation.

Sadat received Begin later this evening for their first round of talks in Mamoura Palace at the tip of a point washed by the Mediterranean. As he met the Egyptian leader's grandchildren, Begin remarked on the synagogue reception, saying: "I have seen the reality of peace today."

Begin and Sadat talked in private while sitting on lawn furniture in the spacious garden of Sadat's residence, then quickly adjourned foor a small dinner with only a few advisers. Sadat's encourage included Vice President Hosni Mubarak, Defense Minister Kamal Hassan Ali and other top aides. Begin, however, brought none of his ministers with him to Alexandria and only one senior adviser, Eliahu Ben Elissar, director general of the prime minister's office.

It appeared unlikely that the two leaders would announce any major departures in their positions on the contested Palestinian utonomy issue after te few hours of talks planned here. Egytian and Israeli officials suggested, however, that Sadat and Begin's private discussions could go a long way toward encouraging cooperation and giving increased momentum to their negotiating teams.

Sadat is know to be eager for gestures from Begins to create a climate in which Palestinians would find it easier to consider participating in the talks. Examples mentioned in Cairo are the release of more Palestinian prisoners in Israel or a softening of Israel's insistence on the right to build new settlements in the occupied West Bank..

For its part, Israel is seeking more concrete normalization in relations between the two countries, For example, although the Egyptian-Israeli border twice has been declared open, little actual moovement has been allowed and there still is not sign of direct air service between Cairo and Tel Aviv.

In principle, the autonomy talks moved ahead last week with the decision fostered by U.S special envoy Robert Strauss to break the discussions into two working groups. The two-group formula, one for the West Bank elections and the other to define the limits of autonomy, got them by an earlier impasse. But it still was only procedural, similar tto laying out pieces on a chessboard without really starting the game.

In the view of Egyptian officials, the talks here, even without spectacular advances, can give Begin an idea of what to expect when the autonomy talks resume Herzliya on Aug. 5.

Begin arrived here in an Israelimade executtive jet to a stiffly formal welcoming ceremony led by Mubarak and other top officials. In a scene reminiscent of previous official visits by Begin, a goose-stepping helmeted honor guard accompanied Begin from his plane tto a reception line as an Egyptian military band played the Israeli and Egyptian national anthems.

As he did during his Cairo visit, Begin vsited a synagogue -- Alexandria's only Jewish house of worship -- where he was greeted by Clement Setten, 74, president of the city's Jewish community of 340 persons, and other Jewish leaders.

Setten, who said 40,000 Jews lived here before the 1967 war, told reporters the mood of the city's 2 million Moslems has changed in the last year.

"I've never had any real problems, but after Sadat's visit to Jerusalem, they don't say 'Salaan ' anymore. They say 'Shalom '" Setten said.

Inside the spacious and stately synagogue, Sarena Mordokh, 80, who was born in Turkey but who has lived here most of her life, said of Begin, "I love him. I would like to kiss him, even if only his hand."

The concregation, for the most part elderly women who seemed visibly moved by Begin's visit, pressed forward for a glimpse of the Israeli prime minister, some of them chanting, "Begin, Begin."

After leading prayers, Begin told the Egyptian Jews, "From the days of antiguity, a Jewish settlement has played a major function in the history of the Jewish people of Egypt. Today you are few but devoted and faithful.

"We have in Israel absorbed many Jews of Egypt. We await the day when you will come to see the land of your fathers and see your relatives and see how we have built a good and beautiful land with God's help."

The narrow street outside was packed with Alexandrians who cheered and chanted "Begin, Begin, Begin" as the prime minister moved to his waiting car.

As the car began to pull away. Begin, to the chargrin of his security detail, ordered the driver to stop and plunged into the midst of the crowd, moving from one side of the street to the other and grasping outstretched hands.

As a squad of troops rushed to push the crowd back, Begin, obviously enjoying the reception, was gently nudged back in the car by his aides for the return to Safa Palace.

Begin has frequently said that the high point was a visit to Cairo last April was a visit to the Cairo Museum when, confronted by a similar but more restrained welcoming, he went into a sidewalk crowd to shake hands of Egyptians. CAPTION: Picture 1, Israeli Prime Minister Menachem Begin waves to a cheering crowd outside a synagogue in Alexandria, Egypt.Begin was in Alexandria to discuss Palestinian autonomy with Egyptian President Anwar Sadat. Ap Picture 2, Egyptian Vice President Hosni Mubarak, left, and Israeli Prime Minister Menachem Begin review an honor guard after Begin's arrival in Alexandria, Egypt. Ap