President Carter finally ventured out on the campaign trail today, bearing a message of optimism that the United States is "turning the tide" on its principal foreign and domestic challenges.

In the first acknowledged campaign appearance of the year, Carter told a noontime rally at the Nationwide Plaza it was time to set the record straight about the nation and its future.

"We do live in a time of challenge, we do live in a time of change, we do live in a time of danger," he told the friendly if not wildly enthusiastic crowd. "But in every area of change, in every area of challenge, in every area of danger, because of our courage and our strength, America is turning the tide."

The president's appearance here included a high school band and red, white and blue balloons, evidence of the careful preparation by a White House advance team that was bent on outdoing its Ronald Reagan counterparts who staged a rally for their candidate on the State Capitol grounds six blocks away.

And it had some typical Carter campaign touches. Halfway through his 7-minute speech in a sunny warm weather, the president shed his suit coat, drawing a cheer from the crowd of about 7,000. Following the speech, on his way to a campaign fund-raiser at a nearby hotel, he sat on the roof of his open-top black presidential limousine to wave to crowds on downtown streets.

Since Carter resumed traveling after the aborted Iranian rescue mission, all of his trips have had heavy political overtones. But today's appearances were unabashedly political and appropriately took place in Ohio, the focal point of the last round of primaries Tuesday and a key battleground in the fall election.

Carter campaign strategists are confident of winning the Ohio primary and predict that this will wrap up the Democratic nomination and offset possible losses to Sen. Edward M. Kennedy (D.-Mass.) in California and New Jersey the same day. The Columbus rally, and campaign appearances by the president later today in Cleveland, were designed to leave nothing to chance.

The president has been speaking in decidely optimistic tones since failure of the rescue mission, asserting that the nation's future remains bright if it sticks with his leadership. That was his central message today to the Columbus rally as he sounded what could be a main fall campaign theme.

"I am not here to say that all of my decisions have been right or popular," Carter said. "I know that they have certainly not all been easy ones. But I will say this: in the last 3 1/2 years we have been tested under fire. We have never ducked or hidden . . .We have made tough decisions and we have taken the heat when we made those decisions. But we have done what was right, and we have always told the truth."

The president, who four years ago campaigned on a pledge to cut defense spending, boated today that the United States "is leading its allies in a steady, sustained growth in defense expenditures." He predicted a decline in the inflation rate this summer and said the country is "on the road to energy security." He made only a fleeting reference to the hostages in Iran, saying the United States had "acted with firmness and with determination against terrorism in Iran."

While Carter was forecasting an improvement in the economy in New York, Charles L. Schultze, chairman of the Council of Economic Advisers, carefully avoided any specific predicitions about how long or deep a recession may be developing. *tThe crowd appeared enthusiastic in seeing Carter, although applause lessened as he recited a series of economic accomplishments under his administration. The president, too, semed happy to be back in a campaign atmosphere but he also did not stray from the posture of an incumbent president somewhat above the fray, not making a single reference to Kennedy or Reagan or their presence in the state.

From Columbus, Carter later today flew to Cleveland where, despite the months away from campaigning, he demonstrated again that he is at his most effective before smaller crowds rather than large rallies.

He spoke first in a sweltering auditorium in surburban Parma, easily fielding a number of questions from friendly members of a senior citizens organization.

Tonight the president spoke from the pulpit of Cleveland's Olivet Institutional Baptist Church, where he asked the congregation to stand and pray silently for Vernon Jordan, the head of the National Urban League who was shot and wounded in Fort Wayne, Ind., earlier today.

Referring to his audience as his "brothers and sisters," Carter used scriptural language to deliver his political message of optimism.

"We have not yet reached the Promised Land, but we're making progress," he said repeatedly.

Obviously enjoying the experience of campaigning again Carter also made the Republican Party sound something like a dread disease. "A president," he declared, "has to face a lot of things -- raging rivers, exploding volcanoes, Republicanism that might come back in November."

The only thing to mar Carter's campaign day was a large protest outside the Cleveland Plaza Hotel, where the president tonight addressed a state Democratic Party fundraiser. The protesters represented a number of causes, but most held signs denouncing rising unemployment and demanding, "Stop Carter layoffs."