U.S. and Soviet negotiators face substantial hurdles in attempting to transform the cooperative spirit of last week's successful meeting between Secretary of State George P. Shultz and Soviet Foreign Minister Eduard Shevardnadze into agreement on a broad range of arms-control issues.

Despite resolving key issues of a treaty to eliminate medium- and short-range weapons, the superpowers remain sharply at odds over reductions in strategic, or long-range, nuclear weapons, which both sides call their top arms-control priority.

They also disagree about many important peripheral issues, such as nuclear testing, despite their agreement Thursday to renew formal negotiations on testing. And as senior officials on both sides indicated in closing news conferences, they also have fundamentally different views about the future direction and ultimate goals of arms control.

For example, Shevardnadze spoke of his hope that settlement of the contentious space weapons and strategic arms debate could eventually enable the superpowers to eliminate "all nuclear arms," an action that the Reagan administration has ruled out.

Shevardnadze also said he hopes that the pending agreement on medium-range and short-range arms will "give moral and political and other kind of support" to a new strategy of "military disarmament in Europe." But U.S. and allied officials, while generally supportive of the pending agreement, strongly oppose any further reductions in European nuclear weapons and have little enthusiasm for reductions in conventional forces.

As Shultz indicated to reporters, the two sides had "a very good exposition of our basic views" on several of these issues, but shifted their positions only a bit. Other U.S. and Soviet officials pointed to the following problems:Strategic Arms: Noting that "We have not yet been able to bring our positions substantially closer . . . on the question of 50 percent reductions of strategic offensive arms," Shevardnadze agreed with Shultz in a joint statement Friday that an "intensive effort should be made to achieve a treaty" in this area "within the framework of the Geneva nuclear and space talks."

But the Reagan administration has yet to accept the Soviet position that strategic arms limitations will be removed if either side deploys a ballistic missile defense in space, as the Pentagon has proposed.

Although Shevardnadze said the Soviets believed this question "is the root question of Soviet-U.S. relations," other U.S. officials cautioned that negotiators on the two sides have yet to agree in any detail on how their current nuclear arsenals of 4,500 strategic weapons should be restructured by a new treaty.

In particular, the Soviets have not agreed to accept what the United States considers an essential limitation on warheads atop land- and sea-based missiles. The United States, meanwhile, has resisted constraints on its submarine and cruise-missile force that the Soviets have emphasized.

U.S. and Soviet officials also agreed that even if the general outline of a U.S.-Soviet deal on strategic arms is in sight, the troublesome topic of verifying compliance with an agreement has hardly been broached.

The Reagan administration's position is that neither side can count long-range mobile missiles deployed by the other side with the precision needed to guarantee mutual security. Some officials hoped a solution to this problem could be worked out in negotiations on mobile missiles to be covered by the medium-range arms treaty, but the issue was not resolved.

Nuclear testing: Although U.S. and Soviet officials hailed an agreement to begin formal negotiations, they were candid about the wide philosophical gap that remains.

"We would have preferred a more radical solution and that is the immediate ending of all nuclear tests," Shevardnadze said. Instead, there will be "stage-by-stage" negotiations that begin with efforts to develop new means of verifying compliance with old treaties that limit the explosive force of nuclear tests to 150 kilotons, equivalent to 150,000 tons of TNT.

The Soviets prefer that the treaties be improved by taking a few measurements of seismic signals near the testing sites in each country. The United States, on the other hand, wants to verify compliance by reading the signals from a cable buried near every blast.

Even if this technical dispute can be settled quickly, the Reagan administration remains strongly opposed to restricting the number or frequency of nuclear tests, as called for in the second "stage" of negotiations specified in the joint statement. The Soviets, however, said they favor a cap of four tests annually with a force of less than one kiloton each.

In the final "stage" of negotiations, the two sides are supposed to pursue "the complete cessation of nuclear testing as part of an effective disarmament process." But Kenneth L. Adelman, director of the Arms Control and Disarmament Agency, said last week that this will be "way, way, way down the road . . . when there's peace on earth and good will towards men." Medium- and short-range missiles: Substantial issues also remain on the topic where the two sides are closest to agreement. Shevardnadze said he hoped for an agreement on the treaty text during Shultz's visit to Moscow late next month.

But the two sides have not completely settled a vigorous dispute over the schedule for withdrawing U.S. and Soviet warheads in Europe, besides the U.S. warheads for 72 West German missiles, where an agreed schedule was developed. The Soviets want the other warheads withdrawn within the first year of the agreement, while the Reagan administration wants the warheads to remain until their missiles and launchers are eliminated.

The Soviets prefer a five-year timetable for this elimination, while the Americans prefer three years.

Finally, the two sides have not yet agreed on detailed measures to verify compliance with the agreement, a topic that will get much scrutiny during administration efforts to obtain Senate ratification next year.