The stock market strode smartly into the New Year.

After a lackluster December, normally the month for price rallies, the Dow Jones Industrial Average rose 7.52 points on the first trading day of 1982.

Nevertheless, trading was light, and the Dow average, which tracks 30 of the biggest and most widely held industrial corporations, performed much better than the stock market as a whole.

For example, the New York Stock Exchange index, which measures the performance of all stocks traded on the nation's biggest exchange, rose 0.09 point to 71.20, a gain of less than 0.1 percent. By contrast, the Dow rose 0.86 percent.

Investors seem eager to buy so-called cyclical stocks, those that will do well as the economy comes out of a recession, said Jerry Hinkle, chief trader for the brokerage firm Sanford C. Bernstein & Co. Most of the major stocks in the Dow average are cyclical in nature.

In the bond market, however, things were not so rosy. Bond prices fell sharply in late trading after the Federal Reserve Board reported that the money supply rose $1.4 billion the week before last. Most analysts had anticipated that the level of checking accounts and cash in circulation would decline slightly.

The key 30-year Treasury bonds lost nearly $30 on a face value of $1,000 today. The Federal Reserve report came out too late to affect stock trading in New York.

Late in the day, reports surfaced that Henry Kaufman, chief economist for Salomon Brothers and one of the most widely respected interest-rate forecasters on Wall Street, told the firm's clients that he expects interest rates to begin to climb again by July. He said the rising rates would occur because heavy federal borrowing to finance the budget deficit would collide with a big spate of corporate borrowing.

Kaufman has been one of the most pessimistic interest-rate prognosticators in the investment community.

Trading volume on the New York exchange was lighter today than on New Year's Eve, which traditionally is a slow day. Nearly 37 million shares of stock were bought and sold on the New York exchange, compared with 40.8 million last Thursday.

The number of stocks that closed higher in price on the New York exchange outnumbered those closing lower by about three to two, with 954 stocks up and 580 stocks down.

"It takes a while for activity to resume, especially among institions," said Hinkle. Institutional traders, such as pension funds and insurance companies, buy and sell stock in big blocks and each year account for a bigger volume of transactions on the major stock exchanges.

On the American Stock Exchange, the index was up 0.78 point to 321.41. Gainers outnumbered losers by 388 to 219.