Mikhail S. Gorbachev opened a disarmament initiative Wednesday - cutting the size of the Red Army and reducing forces in Europe and Asia - then went to lunch with President Reagan and George Bush to seal an extraordinary chapter in U.S.-Soviet relations and lay the groundwork for the Gor-bachev-Bush era.

The Soviet leader, belittling military power as a solution to international tensions, told the U.N. General Assembly that tens of thousands of tanks, as well as troops, would be withdrawn from Eastern European countries and other forces would be pulled from Mongolia and Asia in a reduction in size of the Soviet military machine.And he called for a Jan. 1 cease-fire in Afghanistan, for the creation of a U.N. force to keep the peace there, for the formation of a broad-based Afghan government and for an end to all arms shipments to either side in that country's war. The United States has been aiding anti-communist fighters in that war.

While Gorbachev was spelling out his actions and his ideas, Reagan and Bush flew separately to New York to have lunch with him on Governor's Island, a secure spot across New York Harbor from the Statue of Liberty.

Reagan, Gorbachev and Bush - all three smiling in the bright sunshine and mild temperatures - posed for pictures on the steps of the two-story brick and columned Admiral's House where lunch was to be served. Two cannons of the Civil War era stood outside the building, and the three steps to the front porch were flanked by the American and Soviet flags.

Said Gorbachev, "If we score any points, we can do it only together. If we try to score points alone, nothing good will happen."

Bush let Reagan do the speaking. Responding to reporters, the president reacted cautiously to Gorbachev's speech.

"I think he's sincerely dealing with the problems he has in his own country," Reagan said, adding, "We'll have to wait and see" about the response of the United States and its allies to Gorbachev's proposal for a reduction in forces in Eastern Europe. He said it was something the United States "had been suggesting" for some time.

He said that there have been no meaningful negotiations on conventional-force reductions and that the Soviet leader's proposal "is a unilateral action on their part."

Speaking forcefully at the United Nations for a bit more than an hour, the Soviet leader paraphrased the English poet John Donne - by way of Ernest Hemingway - the Soviet leader said, "The bell of every regional conflict tolls for all of us."

Gorbachev criticized the Reagan administration on one action - its decision to deny a visa to Palestinian leader Yasser Arafat so he could address the U.N. from the same podium Gorbachev was using.

The General Assembly voted to hold a session in Geneva so Arafat could make his speech. Only Israel supported the U.S. action, taken on grounds that Arafat and his Palestine Liberation Organization sponsor terrorism. "We voice our deep regret over the incident," said Gorbachev.

On yet another front, Gorbachev pledged that Jews seeking to leave his country to be re-united with their families will be treated in a "humane spirit." But he said Soviet citizens will still be denied exit visas if they have knowledge of state secrets.

Worldwide, he said, military force no longer "can or must be an instrument of foreign policy."

Urging that international leaders seek a "balance of interests," Gorbachev told a rapt audience of diplomats that nations should "look for ways together to improve the international system and build a new world."

"It is now quite clear that building up military power makes no country omnipotent," Gorbachev declared. "What's more, one-sided reliance on military power ultimately weakens other components of national security.

"Red Alert: City Braces for Gorby Gridlock," screamed the front-page headline of the New York Post.

But New York Mayor Edward Koch cautioned the Soviet leader against too much work and not enough play. "He ought to make this a working vacation," said Koch shortly after Gorbachev arrived. "There is a lot in this city to see. Enjoy yourself, Gorby!"

Gorbachev's visit to New York and the United Nations was the first by a Soviet leader since 1960, when an irate Nikita S. Khrushchev punctuated his rhetorical points by pounding his shoe on a desk in the General Assembly.

One of the few common elements was the English translation provided at the airport by Viktor M. Sukhu-drev, who also translated for Khrushchev 28 years ago.

Wearing a top coat and fedora, and accompanied by his wife, Raisa, and two members of the ruling Politburo, Gorbachev declared: "We believe - and we hope that is the belief of the American side as well - that this meeting will serve the best interests of the United States and the Soviet Union; indeed of all the world."

U.S. officials said they expected spirited discussion of issues ranging from arms control to the lingering Soviet military presence in Afghanistan, but that no formal agreements were likely to result.

Mood was as important as substance at this meeting, with Bush about to take his place as Gorba-chev's equal on the world stage. Bush made it clear that he is willing to work with the Soviets but not about to jump into any serious negotiating while still Reagan's understudy.