What new taxes will Congress choose - and who will pay?

For a weary Congress, drained by the long battle with the now-passed first budget resolution, there will likely be little rest. The coming fight over tax increases could be just as bitter as the budget wrangle.

The delicately crafted budget compromise, passed by razor-thin margins in both chambers, requires Congress to enact $20 billion in new taxes for 1983. Taking the first step in shaping a new tax bill, Senate Finance Committee chairman Bob Dole (R) of Kansas met with Treasury Secretary Donald Regan for two hours on June 23, congressional sources said. A broad outline emerged, with about $10 billion coming from an energy tax, another large hunk from business taxes, and the rest of the $20 billion from scattered miscellaneous provisions.

But in an election year, stepping onto the treeless plain of tax increases makes politicians feel very vulnerable. Filling in the specifics of revenue-raising measures will be a difficult struggle, congressmen and their aides say.

''It looks to me as though we're going to have a difficult time with taxes,'' the ranking House Ways and Means Committee Republican, Barber B. Conable Jr. of New York, told a meeting of the US Chamber of Commerce this week.

The rules of the budget process require the Senate Finance Committee to report its recommendations for tax increases by July 12. In the House, the Ways and Means Committee must produce a list of revenue-raising measures by Aug. 1.

Traditionally, the House committee takes the lead in writing new tax legislation. But chairman Dan Rostenkowski (D) of Illinois, with a later deadline, seems content to let the Senate forge ahead.

''I wouldn't argue with that,'' an aide to Mr. Rostenkowski said when asked if the Ways and Means chairman was lying low and letting the Senate take the political heat.

Rostenkowski says that to ensure even more that all responsibility for tax raises will sit squarely on Republican shoulders, he will not move until the minority members of his committee reach a consensus.

''My chairman is frequently saying to me,'' Representative Conable told the Chamber of Commerce, ''Barber, you let me know when 9 of your 12 Republicans have agreed on what you want to do, and then I'll call a meeting of my 23 Democrats and decide whether we're going to go along or not. (Rostenkowski) not only wants the tail to wag the dog, he wants the dog to walk around backwards.''

With the House in the throes of a political standoff, Senator Dole has become the point man in the congressional effort to increase taxes. He must craft a consensus among committee Republicans, whose ideology ranges from his own relatively moderate views to those of Sen. Steven Symms (R) of Idaho, a committed supply-sider who may bolt if he feels real spending cuts are not forthcoming.

''Sure, it's tough (to reach a consensus),'' says a senior Republican aide on the Finance Committee. ''But I have confidence. We've done it before.''

Senator Dole has refused to tinker with the third year of the tax cut. Originally, he also placed energy taxes off limits. But reaching $20 billion with nothing but ''cats and dogs'' (small tax increases) would require many sharp, time-consuming political battles. Now an energy tax is back on the table, as part of the package reportedly outlined by Mr. Dole and Treasury Secretary Regan.

Any package is also sure to include new taxes for business. A copy of the Finance Committee's latest list of proposed changes contains these possible increases for the corporate sector:

* Some sort of corporate minimum tax, which would raise an estimated $2 billion in 1983.

* A tightening in some depreciation rules, to raise $700 million.

* Accelerated corporate income tax payments, for $1.4 billion.

* Eliminating a law that lets companies with multiyear government contracts defer taxes until the contract is completed. This would raise $1.2 billion.

Other big-ticket changes on the Finance Committee ''laundry list'':

* $4.3 billion in 1983 from passage of a bill to tighten rules on tax compliance.

* $1.5 billion from raising the taxes that life insurance companies pay when they share risk on certain policies.

* $1.2 billion from airport and airway taxes.

* $1.2 billion from a cigarette tax of 8 cents a pack.

* $800 million by disallowing the deduction of state sales taxes.