President Reagan took the campaign for his economic program on the road today with a sharply worded attack on the House Democratic leadership for seeking to sidetrack his latest plan for cutting the budget.

Referring to the leadership's decision to bring up the new budget cuts in piecemeal fashion rather than for a single up-or-down vote, Reagan said: "It's a sad commentary on the state of our opposition when they have to resort to a parliamentary gimmick to thwart the will of the people."

Reagan also took a swipe at House Speaker Thomas P. (Tip) O'Neill Jr. (D-Mass.) for saying that under the Reagan program, "You might wake up one morning and see our country destroyed." Without naming O'Neill, the president said, "Those who make such charges want to return to the same discredited policies that set off America's high fever in the first place."

The president's appearance here before a noisy, friendly national convention of the Junior Chamber of Commerce had all the trappings of a political campaign. The Jaycees greeted the president with repeated shouts of "U.S.A." and "We Want Reagan." Outside, a small group of demonstrators representing labor, poor people and women's groups had their own chant: "We want jobs, not war."

The speech here, the first of three that Reagan will make during a five-day period, is the first major effort since his inauguration to sell his program of budget cuts and income tax reductions out on the road. Reagan had been scheduled to start such a campaign the week he was wounded in an assassination attempt March 30.

That earlier outing turned out to be unnecessary, as the Reagan budget sailed through the House by a 77-vote margin with the help of 63 Democrats. Now, facing a potentially much closer vote on the bill making specific program cuts to conform to the budget, and another close vote on his tax bill, Reagan is seeking to drum up what appears to be slipping support among conservative southern and western Democrats.

After praising Congress for enacting the budget, Reagan said, "Today, I'm asking those remaining committees and the House leadership to join me in going the last mile."

Reagan's attacks on O'Neill and on the House Rules Committee, which has control over the terms of debate on the budget, were inserted into his speech on the plane ride from Andrews Air Force Base to San Antonio. In his prepared text he had been careful not to threaten Congress, but his tune changed when he learned of the Rules Committee's intentions.

The Democrats think the new Reagan cuts will be easier to defeat one by one than all together.

The central message of Reagan's speech was that Congress should not alter his budget cuts or his program for a 25 percent income tax reduction over three years.

"I don't feel I can accept any further changes even though those who are unenthusiastic about tax cuts generally are pressing for a 15 percent cut over two years. . . . I believe the third year is important particularly for small business, which creates 80 percent of our new jobs and which pays the individual tax rather than the corporate tax."

Reagan suggested that the Democrats really didn't want to cut taxes at all.

"If could paraphrase Will Rogers' line about never having met a man he didn't like, it seems that some in government have never met a tax they didn't hike," he said.

The president targeted his criticism at what he called "automatic spending programs" such as food stamps and housing subsidies, which his aides say the House has not cut enough or not cut in lasting ways.

"Without these added reductions we will have given up just as we are on the brink of a great national victory," Reagan said. "Without these reductions we will have nearly $22 billion of red ink, an unbalanced budget more inflationary pressure in the next few years."

He also referred to food stamps and housing subsidies as "good intentions run amok" that would "pile enormous new debt on the backs of our children --heavily mortgaging their future for the sake of temporary social cures." Two of the separate votes the Democrats are contemplating are on the stamps and housing programs.