On a day of record low temperatures, President Reagan yesterday set some notably lofty goals for himself. Key legislators welcomed the bipartisan tone of his second Inaugural Address but warned that tough battles lie ahead on every one of his major objectives.

From elimination of federal deficit spending to sweeping reform of the tax system to the removal of nuclear weapons from the world's arsenals, the agenda Reagan described is as ambitious as any sketched on the 49 previous such occasions.

Despite the applause from the audience of dignitaries in the Capitol Rotunda, some skeptics pronounced Reagan's hopes as artificial as the ceremony -- a reenactment of the oath-taking that took place Sunday at the White House.

From the balanced-budget amendment to the Constitution to a "security shield" against incoming nuclear missiles, the specific objectives the president listed yesterday for his second term are all matters his critics have derided as undesirable, unattainable or both.

But the emphasis Reagan gave them, along with the undeniable fact of his political popularity as he seeks to become the first president in a generation to complete two full terms, guarantee that these few issues will be at the top of the governmental agenda.

None was a surprise. Though Reagan drew rhetorical inspiration from presidents as diverse as John Adams and Thomas Jefferson, borrowing impartially from Lincoln's Emancipation Proclamation and Wilson's New Freedom, the source of most of his ideas could be found in the previous utterances of Ronald Reagan himself.

Any thought that Reagan might redirect his energies -- or revise his doctrine -- as he launched his second term should have been put to rest yesterday. "I do not believe," he said, "you reelected us in 1984 to reverse course."

Former Senate Majority Leader Howard H. Baker Jr. (R-Tenn.) said the speech "emphasized . . . the continuity from the first term to the second . . . . That's what the country expected, that's what the country wanted, and I think that's what the country's going to get."

In his rather brief speech, Reagan returned to several familiar themes -- the need for a continued military buildup, the virtue of shifting federal responsibilities to the states and restricting abortion.

His emphasis on eradicating any symptom of "hatred and prejudice installed in social custom and law" seemed to some listeners to be a response to the latest round of criticism of his first-term policies, just last week, from minority-group leaders. But, in fact, almost identical language, denouncing the "barriers born of bigotry and discrimination," was in the Reagan's first Inaugural. Then, as now, "a healthy, vigorous, growing economy," rather than any specific, remedial federal program, was advanced as the surest cure for inequality.

Reagan's address drew bipartisan praise from members of Congress, who made up a large part of his audience of about 1,000, when the chill temperatures chased the ceremony indoors from the platform that had been constructed on the Capitol's West Front. Such diverse figures as Senate Majority Leader Robert J. Dole (R-Kan.) and freshman Sen. Albert Gore Jr. (D-Tenn.) praised Reagan for invoking the spirit of bipartisanship at the start of what is expected to be a bruising congressional session.

"It was a healing type of speech," Dole said, "and that was good." Gore agreed that "the call for bipartisanship will be welcome in the Capitol and in the country."

But the first test of Congress' cooperativeness and of the president's leadership will come on the issue where Reagan seemed least comfortable yesterday -- the issue of the budget deficits.

In his first Inaugural, Reagan declared: "For decades we have piled deficit upon deficit, mortgaging our future and our children's future for the temporary convenience of the present. To continue this long trend is to guarantee tremendous social, cultural, political and economic upheavals."

In his first term, Reagan did not continue the trend -- he accelerated it. The average annual deficit increased from $48.5 billion in President Jimmy Carter's term to $165 billion under Reagan. So yesterday, Reagan reiterated that "a time of reckoning" has arrived. He said he would submit a budget "aimed at freezing government program spending for the next year," a careful phrasing that leaves room for inevitably higher interest payments on the mounting debt.

But Dole commented that Reagan's promise of a qualified one-year spending freeze fell short of what Congress wants. "We want to get into the next two years," Dole declared, adding that he had warned Secretary of Defense Caspar W. Weinberger in a conversation at the Capitol yesterday that "if we can make them cuts in every other program, they can make them in defense as well."

Dole has pressed Senate Republicans to take the lead in formulating a three-year deficit-reduction plan, without waiting for Reagan's budget submission. Freshman Sen. John D. Rockefeller IV (D-W.Va.) remarked that if he had "abdicated responsibility" on the budget as governor to the extent Reagan had in Washington, "they would have run me out of the state."

Another freshman, Sen. Tom Harkin (D-Iowa), voicing strong doubts about Reagan's rosy description of his first-term accomplishments, said it was up to the Republicans to take the lead on deficits and other problems. "They dug the hole," Harkin said. "Let them say how they'll fill it back up."

Reagan's second major domestic objective, tax simplification and further rate reduction, got only brief mention yesterday, but officials said it will be spotlighted in next month's State of the Union address.

On this issue, Reagan does have a potential of Democratic support, and there are negotiations between the White House and such prominent Democratic "flat-tax" advocates as Sen. Bill Bradley (D-N.J.) and Rep. Richard A. Gephardt (D-Mo.). But the tax-reform effort faces a series of lengthy drafting efforts and congressional hearings. Interest-group opposition to the removal of tax advantages in the present code make its prospects uncertain.

Even greater uncertainty attends the two sides of Reagan's defense and arms-control agenda -- research and development on the controversial antimissile defense system critics call "Star Wars," and negotiation of a treaty with the Soviet Union to reduce offensive nuclear weapons. Sen. Sam Nunn (D-Ga.), an influential member of the Armed Services Committee, said Reagan's plan "is a program that's in bad need of a rational definition," and others warned that the linkage of "Star Wars" and arms control could create more problems than it solves.