Basketball star Bill Bradley and Vietnam veteran Larry Pressler won impressive Senate primary victories yesterday in New Jersey and South Dakota, respectively, as voters went to the polls in eight states on the busiest Tuesday so far in the 1978 political season.

Bradley, 34, overwhelmed five opponents to win the Democratic nomination in New Jersey against Sen. Clifford P. Case (R), 74, who waslocked in an unexpectedly close battle with his 34-year-old conservative challenger, Jeffrey Bell.

Bell was leading Case by a small margin with 90 percent of the votes counted, and one network, ABC, projected the challenger as the winner.

Pressler, 36, a two-term member of the House, won three-fourths of the Republican primary vote in South Dakota, and established himself as the early favorite for the seat of retiring Sen. James D. Abourezk (D). The outcome of the Democratic primary was in doubt.

In Mississippi, where the retirement of Sen. James O. Eastland (D), set off the biggest political scramble in three decades, Rep. Thad Cochran (R) won impressively in the state's first major GOP primary.

The Democratic runoff on June 27 will pit Gov. Cliff Finch, 51, against Maurice Dantlin, 84, who lost to Finch in the 1975 gubernatorial battle but had elements of Eastland's organization supporting him this time.

Ex. Rep. Wayne L. Hays (D), who left Congress two years ago after acknowledging he had placed his mistress, Elizabeth Ray, on the congressional payroll, won his comeback bid for nomination to a seat in the Ohio legislature. He received almost half the votes in a five-man field.

Also in Ohio, Gov. James A. Rhodes (R) and Lt. Gov. Richard F. Celeste (D) won their expected primary victories and will square off in the gubernatorial battle next fall.

The major races yesterday: New Jersey

Bill Bradley, the Rhodes scholar and basketball star making his first effort at elective politics, won an easy victory in the Democratic primary to choose a candidate for the Senate seat held by Republican Clifford P. Case. 74.

But Case was locked in an even battle with Jeffrey Bell, 34, a conservative who complained that Case's liberal voting record over four Senate terms had been unrepresentative of New Jersey's Republican Party.

With 73 percent of the ballots counted, Case and Bell were splitting the Republican vote 50-50.

Bradley, 34, cruised past his chief Democratic opponent, Richard Leone, with 52 percent of the vote of Leone's 35. Alex Menza ran third in the Democratic race with 11 percent.

Since the fall of 1961, when he played his first basketball game for Princeton, Bradley has been a hero to a considerable portion of New Jersey's population.

Thus, when he announced his run for the Senate last fall, his maiden political endeavor, Bradley was one of the best-known individuals in the state.

Leone, a veteran activist in the "reform" wing of the Democratic Party, hired media wizard David Garth to overcome Bradley's formidable advantage in recognition. But then Leone ran into funding troubles and could not buy the lavish media package that Garth had planned for him.

Bradley's defeat of Leone marked a defeat as well for Gov. Brendan Byrne and the state's Democratic Party organization, which worked hard for Leone's nomination.

Bell, a handsome, intense veteran of Ronald Reagan's 1976 presidential campaign set out 13 months ago at what seemed the impossible task of defeating Case, who has been New Jersey's best vote-getter for two decades.

Bell mined a strong conservative streak in the state's Republican Party. Everywhere he went to campaign, he met people who said they would vote for anyone more conservative than Case. Ohio

The stage was set for a battle of generations and political philosophies in the November gubernatorial race, when Gov. James A. Rhodes (R), 68, and Lt. Gov. Richard F. Celeste (D), 40, swept aside opponents.

Rhodes was leading his opponent, state Rep. Charles F. Kurfess, the minority leader of the state House, by 2-to-1 as the votes were tallied.

Celeste enjoyed a 6-to-1 lead over his opponent, Dale Reusch, a Ku Klux Klan leader.

In November, Rhodes will be seeking a fourth term as governor against Celeste - a Kennedyesque liberal from Cleveland. Celeste has been pointing for this race for four years, ever since Rhodes upset Gov. John J. Gilligan (D), leaving Celeste as his obvious challenger.

The Ohio primary also saw the apparent defeat of one Democratic representative and the political revival of a former representative.

Rep. Charles J. Carney (D) of Youngstown, a victim of steel plant shutdowns in his district and negative personal publicity about his acquisition of surplus books from the library of Congress, apparently fell to state Rep. George D. Tablack (D).

Former congressman Wayne L. Hays (D), who left Congress two years ago after acknowledging he had placed his mistress, Elizabeth Ray, on the congressional payroll, won his comeback bid for nomination to a seat in the Ohio legislature. He received almost half the votes in a five-man field. Mississippi

Rep. Thad Cochran swept to a big victory over state Sen. Charles Pickering to win the Republican nomination for the Senate seat being vacated by retiring Democrat James O. Eastland (D).

Cochran's opponent in the general election will not be determined until a runoff election two weeks from now between attorney Maurice Denitin, 48, and Gov. Cliff Finch, 51, who led a crowded field seeking the Democractic senatorial nomination.

With 93 percent of the precincts counted, Dantin had 26 percent of the Democratic vote to 25 percent for Finch. Former lieutenant governor Charles Sullivan and former governor Bill Walker run third and fourth respectively.

Finch's percentage surprised most political professionals in the state, who had expected the governor to lead the Democratic ballot by a wide margin.

Cochran, 40, a handsome, articulate candidate who comes across well on television, is thought to have a good chance to become the first popularly elected Republican senator in the state's history. Iowa

A moderate and a conservative squared off in a close race for the Republican nomination to challenge Iowa's senior senator, Democrat Dick Clark, in November.

Maurie Van Nostrand, 53, a middle-of-the-road candidate who is closely associated with Gov. Robert Ray, and Roger Jepsen, 49, a staunch conservative, waged a rugged ideological contest for the GOP Senate nomination.

Clark, 48, seemed sure to win nomination for a second Senate term. His two Democratic rivals lacked both campaign money and name recognition.

Ray, 49, the popular Republican governor, seemed a shoo-in for the nomination to seek a fifth term. Democrats Jerome Fitzgerald, 36, the majority leader of the state House, and Tom Whitney, 34, supervisor of the state's most populous county, had a tight race to win the formidable assignment of challenging Ray in November. South Dakota

Rep. Larry Pressler took a huge early lead in the Republican primary to choose a candidate for the Senate seat being vacated by Democrat James Abourezk, who is retiring.

Pressler, 36, a flashy campigner who has gained statewide attention during two terms in the House, had 77 percent of the Republican vote over Ron Williamson, with 20 percent of the precincts reporting.

In the Democratic senatorial primary, Don Barnett, 35, the former mayor of Rapid City, was leading Kenneth Stofferahn, 44, a farmer, by a small margin.

National Republican leaders consider Pressler one of their best bets to snare a previously Democratic Senate seat this year.

In gubernatorial primaries, Lt. Gov. Harvey Wollman, 43, took a healthy early lead in a three-way contest for the Democratic nomination, and Attorney General William janklow, 38, was leading the Republican ballot. Montana

Montana officials predicted a large Democratic turnout because of the tight primary battle for the nomination for the Senate seat now held by Paul Hatfield.

Hatfield, 50, who resigned as the state's chief judicial officer in January to be appointed to the Senate seat vacated when Lee Metcalf died, faced a strong challenge for the Democrat Senate nomination from Rep. Max Baucus, 36, who gave up a safe House seat to make the race.

In the three-way contest for the Republican Senate nomination, Larry Williams, 35, an investment counselor who has described his profession as "con artist," seemed the likely winner.

Six Democrats and six Republicans, including Tippy Huntley, widow of NBC newsman Chet Huntley, were seeking nominations for the congressional seat Baucus is vacating. New Mexico

A former governor battled the present lieutenant governor is a close contest to choose the Democratic nominee for governor of New Mexico. The Republicans also had a tight two-way race for the gubernatorial nomination.

Bruce King, 54, who was governor from 1970-74, and Robert Ferguson, 53, who is ending a four-year term as lieutenant governor, were seeking the Democratic nomination to succeed Democratic Gov. Jerry Apodaca, who is prohibited by law from running for a successive term.

Joe Skeen, 50, who lost to Apodaca in 1974, and Albuquerque legislator Bob Grant, 47, were seeking the GOP nomination.

Incumbent Republican Sen. Pete V. Domenici, 46, was unopposed in his race for the nomination for a second term. Toney Anaya, 37, the state attorney general, had no opposition for the Democratic Senate nomination.