WASHINGTON, May 6— ''The budget document is just a political document,'' said Senator Paul Laxalt. ''As soon as we adopt one, it's quickly forgotten.''

It is true that old budgets do not make popular bedtime reading. But the annual ritual of adopting a spending plan for the Federal Government tells more about the politics and priorities of Congress than any other event on the legislative calendar. And this week the unsuccessful efforts of the Senate Republicans to adopt a budget have divided them into three unhappy and unyielding factions.

This fragmentation reflects the evolution of the Republicans from minority to majority status, from outsider to insider. For the last generation, they have been the critics of the legislative process, not the architects of it, and they are not used to the compromise and accommodation that make that process function smoothly.

''It's a different experience for us, so it's been difficult,'' said Senator Richard G. Lugar, Republican of Indiana. Senator Charles E. Grassley, Republican of Iowa, recalled his years as a Congressman and added wistfully, ''It's doggone easy being a member of the minority of the House of Representatives.'' Shift of Power

The Republicans hold 54 Senate seats, so if the Democrats stay united the loss of just five Republican votes could shift the balance of power to a new coalition. Accordingly, both wings of the party have sensed the chance to exert leverage on the leadership by threatening to defect.

The right-wing group, which includes perhaps 10 to 15 senators, has three basic tenets: increasing military spending dramatically; trimming domestic social programs as much as possible and defending the 10 percent income tax cut scheduled for July 1. Only such a cut, they argue, can stimulate the economy, while depriving the Government and Congressional liberals of the ability to spend more money.

The liberal-moderate faction, with five overt adherents and several covert ones, takes the opposite view: hold down military spending increases, protect domestic programs and jettison the tax cut. Otherwise, they maintain, the deficit will swell to monstrous proportions, sending up interest rates and choking off the nation's economic recovery.

The third group, which might be called the pragmatic conservatives, includes the Republican leadership and more than half of the backbenchers. They are searching for a middle ground but have not yet found one. A Campaign War Cry

In the 1980 campaign, these differences were submerged under a single war cry: ''We're not the Democrats.'' And in the first two years of the Reagan Presidency, the Republican legislators were bound together by loyalty to their new leader and pressure from the folks back home.

Those pressures have waned, and many Senate Republicans believe their current disagreements stem from a failure of leadership in the White House. But their criticisms are often contradictory.

Senator Grassley, a pragmatic conservative, accuses President Reagan of charting a ''fuzzy'' course for the party and the public. However, the moderates fault Mr. Reagan for being too rigid, particularly on tax matters.

Many lawmakers say the President has lost much of his ability to enforce party unity by rallying public support for his program. As Senator Dan Quayle, Republican of Indiana, put it, the President ''doesn't have the big consensus or mandate'' behind him that he once did.

In the absence of that consensus, the factions have grown bolder. The conservatives fear the President is drifting toward the middle, and instead of shouting, ''Let Reagan be Reagan,'' their cry is now more like, ''Make Reagan be Reagan.'' Lure of the Middle

The moderates, in turn, feel the Republican Party must move toward the middle to survive. Many of them never shared Mr. Reagan's faith in ''supply side economics'' and in the notion that large tax cuts would produce recovery. Now they fear that continued devotion to that policy will strangle the aging industrial regions that many of them represent.

As Senator Charles McC. Mathias Jr., Republican of Maryland, put it, ''The President is frequently caught between the right wing and the right thing.''

As Senator Laxalt noted, the budget is primarily a political document, and 19 Republican seats are up for re-election. As a result, admitted the Nevada Republican, many lawmakers are ''protecting their political flanks'' by staking out an independent position on the budget.

The moderates also insist that Mr. Reagan and the Republicans must demonstrate a more humane attitude and convince voters that, as Senator Mathias put it, they ''are not just the party of the rich.''

Illustrations: drawing