Even Democrats said yesterday that President Reagan will probably get most of his proposed spending cuts through Congress, and House leaders joined the Senate in putting his plan on a fast track toward passage by midsummer.

There was also agreement among House leaders of both parties to hold up final action on the administration's tax cut until passage of the spending cuts is assured.

House Speaker Thomas P. (Tip) O'Neill Jr. (D-Mass.) and other Democrats protested that the administration still has not submitted crucial details necessary for Congress to act on its spending recommendations.

There were also assorted protests against the spending proposals. Bar associations (on legal services), mortgage bankers (FHA loan guarantees), the president of Amtrak (passenger service outside the Northeast), the governor of North Carolina (Medicaid), all were heard from.

Even so, both houses greeted Reagan's submission of his revised 1982 budget with pledges of cooperation and prompt consideration.

The president is "entitled to his experiment with the American economy" and Democrats will not stand in the way unless it goes too far, said Senate Minority Whip Alan Cranston (D-Calif.) in predicting that Reagan will get "at least 75 percent" of the dollar-volume of the cuts that he is seeking.

Senate Majority Leader Howard H. Baker Jr. (R-Tenn.) was even more optimistic, suggesting that Congress will approve "closer to 90 percent" of what Reagan wants.

Sen. Lowell P. Weicker (R-Conn.) said liberal-to-moderate Republicans like himself could be expected to join Democrats in resisting cuts in programs such as education, housing, job training, health and transportation, but added, "I think the president has the votes. . . . I don't think there's any question about it."

In the House, where Democratic and Republican leaders reached a rare joint agreement over handling of the budget, Budget Committee Chairman James R. Jones (D-Okla.) said, "$25 billion to $30 billion is certainly a realistic figure, and it could go higher."

Republicans, meanwhile, introduced the Reagan tax bill in both houses.

The joint agreement on scheduling for the House anticipates final action on program cuts for fiscal 1982 by July 15, with approval even sooner for a package of $16.5 billion I rescissions and deferrals for money set up to be spent before the 1981 fiscal year ends Sept. 30.

House Republicans had been pushing for quick action on the 1981 cuts, but couldn't push too hard in light of the fact that Reagan has yet to submit most of his specific 1981 spending reduction proposals to Congress.

For their part, O'Neill and some other Democrats had been complaining about overly hasty treatment of the economic package, but would up agreeing on a schedule that, at its windup at least, differs little from the timetable set by the Senate Republican majority.

The Senate schedule also calls for completion of the spending cuts by mid-July, but the Senate plans to start more swiftly, beginning with Senate Budget Committee mark-up of program reduction instructions to varius legislative committees next week.

Baker, in what he described as "breakneck speed," said the Senate expects to pass the instructions by April 15, maybe even by April 1. The House, tying the cuts to the 1982 budget resolution and following a more deliberative course, plans to pass its committee instructions by the end of April.

Under this so-called "reconciliation" process, legislative committees are instructed by both houses, based on recommendations from their budget committees, to make overall dollar savings in programs under their jurisdictions.

The committees are given a deadline to come up with specific savings, and these savings are then enacted in a package by Congress. The process erodes traditional committee powers, and some committees have already balked at cuts as deep as Reagan wants.

Even the Republican-controlled Senate Labor and Human Resources Committee on Monday chipped $2 billion off the $11 billion in cuts that Reagan proposed for various health, education, jobs and social service programs, but Baker predicted yesterday that Reagan will win in the end.

Senate Budget Committee Chairman Pete V. Domenici (R-N.M.) was quoted as telling Republicans yesterday that he would resist all suchh erosions of the Reagan package.

Democratic complaints about missing links in the Reagan package included the absence of specifics on proposed 1981 cuts and what O'Neill described as incomplete information on how Reagan would merge a number of catagorical grants into general block grants, giving state and local governments more flexibility over spending while the overall amount of spending is reduced.

"Nor has the administration yet provided any greater analysis of the impacts of their proposed cuts. Who will be affected? How will changes in one program affect another program?" O'Neill asked.

After introducing the tax package, Sen. Bob Dole (R-Kan.), chairman of the Finance Committee, said the underlying philosophy of the administration's proposal will win strong support among members of his panel.

Dole said he has been reluctant about a full commitment to individual tax reductions of 30 percent over three years because substantial spending cuts and continued fiscal responsibility are necessary ingredients in the Reagan program.

"My reservations have been based upon a concern that, if we enacted a tax cut before spending cuts were acted upon, there would be little assurance that the spending limitations would ever be enacted," he said.

Dole said it would be unrealistic to assume that the Finance Committee will not modify specific provisions before sending Reagan's package to the Senate floor.