President Reagan said yesterday that the superpower agreement in Geneva this week could be "the beginning of a new dialogue" with the Soviet Union and asked Moscow to do its part to bring "new life and positive results" to a reinvigorated arms control process.

There were signs, though, that one important facet of the U.S.-Soviet agreement -- involving future talks on outer space issues -- was not nailed down to the extent that officials suggested in first reports from Geneva. State Department officials said a running controversy over the scope of the space negotiations was not completely resolved by Secretary of State George P. Shultz and Soviet Foreign Minister Andrei A. Gromyko, leaving a touchy issue as an early order of business when substantive bargaining begins later this year.

That bargaining, under the terms of the Tuesday agreement, is to involve long- and intermediate-range offensive weapons, on which talks were broken off last year, as well as preventing an arms race in space.

Reagan's comments were a highlight of a day in which senior U.S. officials fanned out through Western Europe and other parts of the world to brief key U.S. allies on the agreement with the Soviets.

Gromyko, meanwhile, left Geneva for home with expressions strikingly similar to Reagan's. The Soviet diplomat called the agreement setting the stage for expanded arms talks "a certain step in establishing a dialogue" between the two countries but spoke of "the immense tasks" that lie ahead in the course of the future negotiations.

In opening his news conference last night, Reagan reminded his audience that "our differences with the Soviet Union are many and profound." He said the new U.S.-Soviet arms negotiations "will be difficult" but "we will persevere."

The president said, "It is my hope that this week's meeting in Geneva, while only a single step, is the beginning of a new dialogue between the United States and the Soviet Union."

Saying that the United States will be "flexible, patient and determined" in the forthcoming negotiations, Reagan also declared, "We now look to the Soviet Union to help give new life and positive results to that process of dialogue."

Reagan referred to the agreements to restart the arms control negotiations, one of the few diplomatic achievements of his time in office, as "American diplomacy at its best."

He paid tribute to Shultz, national security affairs adviser Robert C. McFarlane and other members of the large and diverse U.S. team at Geneva. The chief executive maintained, despite reports to the contrary, that "there has been no infighting" in preparing the negotiating position for Geneva.

Reagan defended his Strategic Defense Initiative (SDI), as a research program in its early stages that is a long way from what is implied by televised animations of "Star Wars" against enemy missiles in space. "Star Wars" will be on the table in the "new" negotiations set up by Shultz and Gromyko, but the United States would be "way ahead of ourselves" to consider setting limits on such a research program, Reagan said.

Reagan pledged that international negotiations would be undertaken when and if the futuristic defense mechanisms are ready for deployment. This probably would be necessary because of existing limitations imposed by the 1972 U.S.-Soviet Antiballistic Missile (ABM) Treaty.

Resumed negotiations on reductions in strategic arms, as contemplated in this week's U.S.-Soviet talks, are likely to be complicated by the politically sensitive problems of verifying Soviet compliance, the president acknowledged.

Charging that the Soviets "have had a past record of violating agreements," Reagan said that "verification to the extent possible is going to be a very necessary feature in our negotiations." At the same time he acknowledged that "absolute verification is impossible."

In response to questions, Reagan indicated a willingness to continue taking some older U.S. nuclear missiles out of service to stay in compliance with the limitations under SALT I and SALT II strategic arms agreements as new missile submarines are added to the U.S. force. The administration has said it will not undercut these limits, although the SALT II agreement was never ratified, as long as the Soviets do not undercut them either.

Reagan's comments came several hours after a personal briefing on the talks by Shultz, who flew home early yesterday. A White House official quoted Shultz as saying in an Oval Office meeting, which was also attended by Vice President Bush, that "we got what we wanted."

The most difficult and contentious issue in the 14 1/2 hours of Shultz-Gromyko talks, by the account of nearly all U.S. officials on scene, was the question of negotiations about military aspects of space. The Soviets, who have reacted sharply against Star Wars, argued in Geneva that space negotiations should be structured to prevent this potential new U.S. use of the heavens to intercept and destroy enemy missiles.

The United States position is that such an antimissile program in space would be defensive in character, could serve in the long run to stop the arms race and should not be limited or restricted, especially since it is still in the research stage.

This continuing disagreement was the focus of down-to-the-wire bargaining between Shultz and Gromyko late Tuesday, according to State Department sources. The Soviet side wanted the space negotiations to focus on limiting Star Wars, which it considers "offensive," on grounds that if one side looked likely to build a workable space defense, the other might be more tempted to launch a first strike while its weapons still might enable it to succeed. The United States wished to focus the discussion on "defensive" weapons in a way that includes Soviet land-based ABM facilities.

The final outcome was an agreement by the two sides to finesse the issue by not describing in their joint communique exactly what the new space arms negotiations are to cover, instead describing the objective only in the most general phrase, "preventing an arms race in space."

In remarks to reporters that he termed "my own views," Shultz said the space negotiations would address "space arms, whether based or targeted on Earth or in space." To the U.S. side this clearly includes the Soviet ABM developments. The sources said the Soviets did not explicitly agree to Shultz's description, though they did not specifically object to it.

How the space negotiations are precisely defined is likely to be an early issue for the U.S. and Soviet teams when they finally sit down to bargain in a month or two, the sources said.

Tuesday's agreement did not specify a time or place for the new talks, saying this is to be decided within a month. "The idea of March as the time was in the air" but was not seriously addressed, an official said. Another source said Geneva is the likely site.

Officials said the Soviets apparently will appoint a separate chief negotiator for each of the three related negotiating "groups" to be formed on strategic offensive arms, intermediate-range missiles in Europe, and space issues. One of these, probably strategic arms negotiator Viktor P. Karpov, is likely to be overall chief of the Soviet delegation, the sources said.

On the U.S. side, officials said there has been no decision about the identity or even the number of the chief U.S. negotiators. Reports that Washington attorney Max Kampelman is to be named overall chief negotiator are premature, officials said. But they acknowledged that Kampelman is one of those Shultz is considering for the job.