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The battle over the budget now begins in earnest on Capitol Hill, but the atmosphere is more like a carnival, with each faction hawking its own version of how to collect and spend federal dollars.

At packed meetings throughout the US Capitol, groups from left, right, and center explain their proposals, trying to woo either public opinion or enough members of Congress for a victory.

''I still think the Budget Committee's is best,'' says US Rep. James R. Jones (D) of Oklahoma, who chairs the House Budget Committee. His ''Jones budget'' is the rough equivalent of a Democratic budget, even if a number of Democrats have rejected it.

''We think we've got the most compatible set of numbers,'' says a House Republican leadership aide about the Republican plan, christened the ''bipartisan recovery budget'' since it has support from some conservative Democrats.

Smaller factions are calling for more money for student aid, job training, and other social programs. One House member is holding out for more money for the military.

About the only general agreement this week is that Congress must pass a budget soon or else face dire consequences in the economy.

At virtually every budget event, at least one member points to the need to prove to financial markets that Congress can control spending. Unless they are convinced, the reasoning goes, interest rates will stay high and business growth be stunted.

So with the urgency of the recession facing them, the Senate is expected to pass its budget resolution this week, and the House will follow early next week. A joint conference will work out the differences, although there is little hope now that work can be completed by Memorial Day. The budget action is already well behind the May 15 official deadline.

Congress has already pushed aside the major stumbling block toward a budget agreement by deciding not to deal with social security benefits, at least until a blue-ribbon panel reports its findings next fall.

But still at issue:

* Taxes. Democrats and liberals, as a rule, want to raise more taxes. Although their budget proposals do not outline which taxes would go up, these groups focus sharply on the third year of the Reagan personal income tax cut, which could be postponed, reduced, or eliminated.

The Democratic-Jones budget calls for $676 billion in federal revenues compared with $665.3 billion under the House Republican plan. On the Senate side , budget chairman Pete V. Domenici (R) of New Mexico asks for $668.2 billion in revenues.

* Deficits. Virtually all of the budgets floating around Capitol Hill have projected 1983 deficits hovering close to $100 billion.

One of the highest, however, is the Republican Senate budget committee deficit, which is $115.3 billion. Although the figure began lower, it grew by almost $10 billion when Senator Domenici agreed to drop projected ''savings'' in social security and to add money to other programs such as student loans, housing, and medicare.

The lowest deficit figure, projected by a coalition of moderate Democrats and Republicans in the House, is $95 billion.

* Military. Spending for defense will go up substantially no matter whose name goes on the budget, and the only question is how much. For 1983 outlays, only a hairline separates the major budget plans, with the Domenici plan offering the most, $215.3 billion, the House Republicans $213.2 billion, and the Jones plan $212.6 billion.

* Domestic programs. Democratic and liberal plans would spend more on social programs and Republicans would spend less, although they have had to yield to some social spending to attract moderate ''gypsy moth'' members.

As the factions prepare for the last rounds of the debate, one unanswered question is whether Reagan can again cast a shadow over Capitol Hill and win a victory in the nominally Democratic House. Without some pulling and tugging, no group has the votes to pass a budget in the House.

US Rep. Silvio O. Conte (R) of Massachusetts predicts it will take ''a lot of talking and massaging . . . and (White House) tie clasps and stick pins'' to produce a majority for the Republican budget.