Goading the Democratic House to speed up work on taxes, the Senate Finance Committee yesterday began to mark up its own bill and promptly approved in principle, by 12 to 2, President Reagan's proposed three-year, 25 percent cut in individual income tax rates.

Committee Chairman Robert J. Dole (R-Kan.) called the vote "significant" and promised to send a complete bill to the Senate floor by next Thursday. The committee also voted, 17 to 0, to keep the cost of its bill within the totals proposed by the administration: $2.1 billion this year, $38 billion 1982, $93.4 billion in 1983, and $149.6 billion in 1984.

Meanwhile, the House Ways and Means Ways and Means Committee tentatively completed action on the business portion of its tax bill, which would provide as large a cut as Reagan proposed and in the process fundamentally structure the way businesses can write off investments in equipment. The committee, over the heated opposition of most Republican members, also approved a proposal under which companies in six "distressed" industries -- airlines, automobile manufacturing, mining, paper, railroads and steel -- could claim $3.3 billion worth of unused past investment tax credits. The Democrats advanced this as a way of providing troubled companies such as Chrysler Corp. with cash; provided they used the money to buy new equipment, such companies would be entitled to refunds of some of their past taxes paid.

Rep. W. Henson Moore (R-La.) called the retroactive investment credit provision, which carried 17 to 10, "yet another bailout," and he added, "You guys have come a long way toward being Republicans on taxes, but this goes too . . . far."

The Finance Committee has reached broad agreement with the administration on 14 items for inclusion in a bill that would track closely the revised Reagan plan, Dole said. Five other items, including indexing individual taxes for inflation after 1984, some additional savings incentives and help for unprofitable industries, are still being negotiated.

Dole said the committee also plans to write a provision designed to crack down on so-called commodity tax straddles, which would reduce the net cost of its bill. The House is expected to adopt a similar provision.

Usually the Senate waits for the House to originate major tax bills, but Ways and Means, with Democrats in charge, has been moving more slowly than the Reagan administration wants. But Reagan has called upon Congress to send him a tax bill by August, and yesterday's start in the Republican-run Finance Committee was an effort to pressure the Democrats to comply.

"We've been waiting for the House to produce a tax bill for a year and a half," Dole said, as he urged the committee to move swiftly to ensure "some tax relief in 1981, partly as a gesture of good faith to the American taxpayer and partly to improve the climate for business expansion immediately." tLast year the Finance Committee reported out a tax cut bill, while the House refused to move on one.

Two Democrats on Finance, Sens. Bill Bradley of New Jersey and George J. Mitchell of Maine, voted against the administration's plan for a 5 percent cut in personal tax rates in October, followed by 10 percent cuts in July 1982 and 1983. Bradley said he planned to offer at least two amendments to the committee next week to tilt the cut in favor of middle- and lower-income groups, and to make the state of the economy a condition of the third-year cut.

Three of the 12 voting in favor of Reagan's individual cuts were Democrats, and Dole indicated to reporters that he expected to get the Reagan alternative through the committee and the Senate despite some Democratic opposition.

Ways and Means completed work on a new system of business taxation that would let companies write off the entire cost of an investment in the year it is made rather than depreciate it over a period of years. In addition the Democrats supported future cuts in the corporate tax rate from 46 percent to 34 percent by 1987.

Beginning this year, any business would be able to "expense" the first $25,000 worth of investments in equipment, a provision that would cover all such investments by about 86 percent of all businesses. For higher levels of investments, the new approach would be phased in over five years to hold down the revenue loss.