President Reagan returned to the White House yesterday to begin the second chapter of his presidency, in which new characters and problems promise to be introduced at the same time that Reagan does battle yet again to cut the federal budget.

His month-long California sojourn marked the end of the single-minded and remarkably successful opening chapter, in which Reagan won passage of the budget-cutting and tax-limiting bills. The open question is whether it also marked the end of his honeymoon.

The nation is no less restless over high interest rates and inflation than it was when Reagan started his vacation, but his efforts to defend and explain his economic programs will now be complicated by competing demands for action on an array of foreign policy and domestic problems.

This other agenda, deferred at White House initiative to leave the decks clear for the congressional battles over budget cuts and taxes, includes the administration's plan to sell sophisticated radar planes to Saudi Arabia, a decision on how to base the MX missile and what manned bomber to build, cuts in Social Security, a farm-bill struggle on Capitol Hill, the extension of the Voting Rights Act and the so-called "social issues"--such as school prayer and anti-abortion efforts--dear to the conservative wing of the Republican Party.

The sense that success becomes harder to find as urgent and competing problems press for presidential attention could have been written about toward the end of the first year of any recent presidency. It is a problem that the Reagan White House is well aware of, and one the president's aides hope will be overcome by Reagan's ability to rally support for his programs.

"The achievements of the first six months surprised us," one White House official said this week. "Now the real work starts."

It begins as, at home, minorities, the poor and interest groups representing them are bracing for the first real impact of the initial Reagan budget cuts.

Abroad--in Latin America, Africa and Europe--there are almost daily expressions of dismay over one or another administration policy.

Friends as well as foes of the United States have criticized the decision to build and stockpile neutron bombs, the go-slow approach to strategic arms control and the decision to take a more conciliatory attitude toward South Africa.

For example, Mexican President Jose Lopez Portillo, with whom Reagan has formed a good personal relationship, condemned the neutron bomb as "the worst that mankind has produced," and reiterated his differences with Reagan over El Salvador this week.

Another complication is that Reagan's relaxed presidential work habits have come in for more open comment than ever before, accompanied by signs of strain among his "Big Three" advisers--counselor Edwin Meese III, chief of staff James A. Baker III and deputy chief of staff Michael K. Deaver.

The three took turns being with the president during the California month, and Meese and Baker, each in his own way, made certain he was visibly exercising authority during his stretch of duty.

Meese's public image took a beating after he decided not to awaken Reagan with news of the dogfight with Libyan planes, and Meese had the highest possible public profile during Reagan's visit to an aircraft carrier.

When Baker arrived to relieve Meese, articles appeared in the media critical of Meese and suggesting he may be too fond of the limelight.

Reagan was asked last week in California how he liked being president. "After all the horror stories about the job, I'm kind of enjoying myself," he replied.

Criticisms that one of the reasons he enjoys the job is that he spends only a few hours a day working could damage Reagan's standing, but some of his top advisers believe that in the Reagan presidency "less can be more."

In their view, the most effective way Reagan can govern is to save his personal intervention for the major battles in order not to dilute the impact he can have when he does personally take a hand.

He returned tan and rested, with his approval rating in the polls hovering around a healthy 70 percent, bolstered by his good fortune in picking up three enemies whose public support is minute at best: Libya, North Korea and the air traffic controllers' union, PATCO.

Reagan has another important political asset going for him that is also something over which he has little, if any, control. No one is leading the opposition. Thus, if Reagan is criticized or seen to have fumbled on some issue, there is no political figure stitching those events into a pattern and providing a rallying point for potential opponents.

After a private weekend, the president will open his new chapter Monday when he travels to New York to make a symbolic presentation of $87 million in federal funds to build a new West Side highway in Manhattan. It will be the fulfillment of a campaign pledge popular with New York labor unions that expect to get thousands of construction jobs from the project.

On Tuesday and Wednesday Israeli Prime Minister Menachem Begin will pay his first visit to Reagan. Later in the year Prince Fahd of Saudi Arabia and King Hussein of Jordan are to visit the White House.

Before the president began his vacation, Meese listed crime, narcotics control, transfer of some powers to state and local governments, and urban economic development as the most likely issues Reagan would want to take up in chapter two of his presidency.

However many new ones he adds, the last month has made clear that the battle to control the federal budget will be a major part of the drama.

On the one hand, this is good news for the administration because the Reagan White House has proved it is skilled at fighting and winning congressional bouts over the budget, and reducing the size of federal government remains a popular issue around the nation.

At the same time, it is unlikely that the White House can focus such complete attention on the issue now as it did in the first seven months of the year, and the need for further cuts can be portrayed as evidence that the administration's first estimates of what was needed were unrealistic.

"We have to make the economic thing work," said a White House official. "That means a major effort will have to be made on the fiscal 1983 and 1984 budgets."

The slump in the stock market and the continuation of high interest rates have given White House officials a sense of urgency in demonstrating their continued vigilance against swelling budget deficits.

In spite of their full plate of problems, the president and his men returned to Washington confident that things are still going their way and that taking August in California did not have a political price.

"We feel the momentum is still there, notwithstanding that the president had a well-earned vacation," one official said.