President Reagan begins his second term today from a peak of public approval higher than at any time since the flood of sympathy after he was shot in March 1981, and with large majorities expecting continued progress in the economy and relations with the Soviet Union over the next four years.

His party also is strengthened, to the point where about as many people consider themselves Republicans as Democrats now.

These situations prevail even though majorities question many of Reagan's specific policies. They doubt that he will do better by the poor in his second term than in the first, and doubt that he will be able to deliver on his promise to roll back the federal budget deficit. Most people also expect income taxes to be raised substantially during his second term -- a move he has said he opposes and they overwhelmingly do. A majority also thinks the United States will become more deeply involved militarily in Central America.

These are among the chief findings of a Washington Post-ABC News poll, conducted Jan. 11 to Jan. 16, in which people were asked both to rate Reagan's performance so far and to predict how he will do in the future.

In the poll, 68 percent said they approve of Reagan's handling of the presidency, and 28 percent said they disapprove. That is his highest popularity rating in almost four years.

The approval rating is only 5 points below Dwight D. Eisenhower's in a Gallup poll as he took office a second time in 1957, and sharply higher than Richard M. Nixon's at a similar point in 1973. The tendency of men more than women to rate Reagan favorably still exists, but not in terms that hold much promise for Democrats. Among men, 75 percent rate the president favorably, among women, 62 percent.

Blacks, the segment of the electorate most opposed to Reagan, give him a higher rating than at any previous time, with 37 percent saying they approve his handling of the presidency. That is slightly better than the rating blacks gave Reagan when he first took office, and more than double his standing among them through most of the first term.

By 59 to 36 percent, those interviewed said that in general the country is going in the right direction. That optimism level is higher than in any other Post-ABC News survey during the past four years.

Like other recent surveys, the Post-ABC poll shows the Republican Party rapidly gaining ground.

Thirty-four percent of those interviewed said they were Democrats, 29 percent Republicans and 36 percent independent. Not since the early 1950s has the GOP come so close to the Democrats. As recently as 1978, polls found twice as many declared Democrats as Republicans.

Perhaps more important, when independents are apportioned according to their leanings (meaning the way they tend to vote) the Republicans edge past the Democrats. In that respect, 46 percent of the people are Republicans or lean that way, 45 percent are in the Democratic column, and 9 percent are independents who do not lean.

By 49 to 33 percent, people interviewed saw the Republicans as better able than the Democrats to cope with the nation's problems over the next few years.

Until recently, Republicans always came in second on that question; as recently as a year ago, the Democrats were ahead, 42 to 37 percent.

A majority of Americans has also adopted one of Reagan's main themes, that government should do less about national problems, leaving individuals and business to do more. Fifty-seven percent in the survey said the government should continue to do less, while 38 percent said more.

People were asked in the poll to give Reagan a grade of A, B, C, D or F for his handling of a variety of issues in his first term, including the economy and inflation, the deficit, reducing chances of nuclear war, reducing poverty, and handling toxic wastes and other environmental problems.

Only on the economy and inflation was Reagan given an average grade as high as B. On each of the others, he got C or C+. But two-thirds of those who gave him an average C grade on all issues combined also voted for him as president over Walter F. Mondale last November.

Looking ahead, a majority of about 2 to 1 expects Reagan to make substantial further progress in improving the economy and keeping inflation under control, and, by 66 to 28 percent, those interviewed also expect him to make substantial progress in dealing with the Soviet Union.

A 56 percent majority expects government action that will make it harder for women to obtain abortions, and 51 percent say they feel there will be movement toward official periods of voluntary prayer in the public schools.

Most people interviewed, 57 percent, expect the country to avoid a recession in the next four years, but 56 percent are pessimistic about seeing a major reduction in the deficit.

On deficit-reducing issues, the public is vehemently against making substantial cuts in Social Security (94 percent opposed) or substantially increasing taxes (79 percent opposed).

But despite Reagan's pledge not to do either, 42 percent said they expect Social Security benefits to be reduced substantially and 67 percent said they expect a significant tax increase.

Similarly, by 62 to 35 percent, the people interviewed opposed substantial cuts in social programs to cope with the deficit -- but by 65 to 32 percent they said they expect such cuts to be made.

The greatest support for cuts was found in the defense budget, with 46 percent saying military spending should be reduced substantially. That, however, was the area in which the fewest people -- 23 percent -- thought substantial cuts would actually be made.