The House yesterday wrapped up its budget cuts for next year with a major restructuring of dairy and grain price support programs, and House-Senate conferees tentatively approved $12.5 billion in cuts over the next three years in the Medicare, Medicaid and welfare programs.

But in a day of budget-cutting elsewhere on Capitol Hill, the Senate gave all but final approval to the administration's proposal for $355 million in increased economic aid for Caribbean Basin and Central American countries.

And the Senate also approved more than $700,000 to complete a new gymnasium for itself. With the administration-blessed three-year tax increase of nearly $100 billion remaining the biggest obstacle to completion of Congress' summer budget business, these were the major developments yesterday:

* The House approved what Democrats claimed would be $4.6 billion in cuts in the farm and the food stamp programs over the next three years, including a new system of dairy price supports and government payments to grain farmers for not planting.

Republicans were skeptical of the savings claims.

The House voted to cut only about half as much from food stamps as the $2.5 billion over three years the Senate voted last week, and beat back Republican efforts to come closer to the Senate bill.

* House-Senate conferees tentatively agreed on $12.5 billion in cuts over three years from Medicare for the elderly and Medicaid and welfare for the poor, largely from new limits on what the government would pay hospitals and doctors under these programs and on how much payments to hospitals could rise each year.

* The same set of conferees wrestled inconclusively with the proposed three-year, $98.9 billion tax increase bill as President Reagan continued, in a series of meetings at the White House, to press rebellious and wavering Republicans to support the measure.

* The Senate, voting 54 to 42, dismissed a procedural objection from Democrats to inclusion of $355 million in Caribbean economic aid as part of a big supplemental appropriations bill for the rest of fiscal 1982. It also rejected, 55 to 40, a proposal to reduce the Caribbean outlay by half, leaving a final vote on the issue until today. The money had been left out of the House version of the bill on grounds that the program had never been authorized. The economic aid also included $128 million for El Salvador, an amount that has been steadily resisted by the House.

* As the Senate began work on the House-passed $14 billion supplemental money bill, it also rejected -- somewhat squeamishly -- a proposal to save $736,400 by eliminating funds for a third Senate gymnasium. The proposal to forgo the gym, advanced by Sen. William Proxmire (D-Wis.), was defeated, 50 to 48. "We won't stop until every senator has his own building," protested Proxmire, claiming there was no end to what the Senate, despite its willingness to cut other people's programs, would spend on itself.

While House passage of the farm and food stamps savings measure completed House action on spending cuts mandated by the budget resolution that Congress approved earlier this summer, Democrats and Republicans were at odds over whether the cuts were sufficient to meet the budget targets.

Even though the House spurned savings of $5 billion over three years by refusing to cap annual cost-of-living increases for federal workers' pensions, Democrats contended that the House, through other spending cuts and revenue-raising measures, met and even exceeded its three-year target of $27.1 billion in deficit reductions. But Republicans claimed it fell short by $10.4 billion.

Republicans said the House Ways and Means Committee claimed savings of nearly $19 billion that included more than $11 billion in tax increases rather than spending cuts, falling more than $6 billion short of its target of $13.8 billion for spending reductions. Some other committees failed as well to meet their individual targets, the Republicans claimed.

But House Budget Committee member Leon E. Panetta (D-Calif.) claimed that the bottom-line deficit reductions were $29.3 billion, or $2.2 billion more than the House had been ordered to save. Panetta said his claim was backed up by the Congressional Budget Office.

The House and Senate are already in conference over the Ways and Means share of the budget cuts, including the Medicare and Medicaid provisions tentatively agreed on yesterday. Now they will go to conference on the rest of the savings, which total $12.6 billion from the Senate and less from the House because it, unlike the Senate, did not approve the federal pension limitations.

The Democrats say this final package, including housing and veterans as well as food-related program cuts, amounts to $9.3 billion, while Republicans say it adds up to no more than $7.1 billion.

Panetta and the Republicans agreed that the cost-of-living raises, farm programs and food stamps will be the key issues in conference.

In the debate over Caribbean aid, Senate Democrats tried to eliminate the entire package from the supplemental money bill, pointing out that the $355 million had never been authorized by a Senate committee.

Sen. Patrick J. Leahy (D-Vt.), normally a foreign aid supporter, said the Senate was being asked to vote a "blank check" for Caribbean nations. In times when the administration is cutting back food stamps, housing and other domestic programs "we should at least ask what this money is for," he said.

The biggest part of the package is $128 million for El Salvador, mainly to help that beleaguered country solve its balance of payments problems. Sen. Christopher J. Dodd (D-Conn.) claimed the money would go to "subsidize the capital flight" out of El Salvador.

The House two weeks ago rejected the Caribbean package on a point of order because there was no authorization for it. The administration then put on a quick drive to get the measure to the Senate floor through the Appropriations Committee, bypassing the Foreign Relations Committee, which already had reported out a drastically different version the administration opposes.

At a meeting at the White House yesterday, Senate Appropriations Committee Chairman Mark O. Hatfield (R-Ore.) reportedly balked at Reagan's objections to some other items in the omnibus money bill, contending that the administration itself has been responsible for spending add-ons, such as the Caribbean money.

Republican sources said senators present at the meeting with Reagan came away with the impression that he may veto the measure, which includes at least $624 million in domestic spending that he doesn't want, including funds for education for the handicapped, jobs for the elderly and aid for poor college students. The administration also objects to the fact that neither the House nor the Senate included about $2 billion in additional defense spending that the administration wants.

The major "must" item in the measure is funds to cover military and civilian government pay raises that have been paid since last October. But Senate sources said the money is not actually needed until late next month, meaning there is time to work out problems with the administration on the measure.

In an opening statement on the bill yesterday, Hatfield said he would attempt to override a presidential veto.