Rep. Peter W. Rodino Jr., who presided over the impeachment hearings of former president Richard M. Nixon, easily defeated a black challenger in his Democratic primary in New Jersey last night, while Sen. James Abdnor defeated Gov. William J. Janklow in the South Dakota Republican Senate primary.

In early returns in California, Republican Rep. Edwin V.W. Zschau held a slight lead over Los Angeles television commentator Bruce Herschensohn in their battle for the right to take on Democratic Sen. Alan Cranston. With 37 percent of the vote in, Zschau had 31 percent and Herschensohn had 29 percent with the other 11 candidates far behind.

In another statewide race of broad interest, Proposition 51, the so-called "Deep Pockets" initiative that would limit liability claims in multi-defendant lawsuits, appeared to be passing by a wide margin. Backers contended that it would help local governments and insurance companies while opponents argued that it would harm injury victims and consumers.

With 85 percent of the precincts reporting in New Jersey, Rodino, 76 and seeking his 20th term in Congress, had 59 percent of the vote. Newark City Councilman Donald M. Payne, 49, who sought to be the state's first black congressman, had 36 percent with two other candidates trailing far behind.

In South Dakota, with 95 percent of the precincts reporting, Abdnor had 54 percent and Janklow had 46 percent of the vote.

Yesterday was the biggest primary day of the year so far. The Republican Senate races in South Dakota and California could affect GOP chances of retaining control of that chamber. Democrats see Abdnor's seat as one of the Republicans' most vulnerable, while GOP capture of Cranston's seat would be a severe setback to the Democrats' hopes of regaining control.

In the South Dakota Democratic gubernatorial primary, state Rep. Lars Herseth upset former governor Richard Kneip, who was heavily favored. With 79 percent of the vote counted, Herseth had 45 percent, Kneip had 37 percent and Public Utilities Commissioner Ken Sofferahn had 18 percent.

In the South Dakota Republican gubernatorial primary, former state House speaker George Mickelson had 35 percent and former representative Clint Roberts had 32 percent with two other candidates trailing. There will be a runoff June 17 if one candidate does not win at least 35 percent.

In another of the nine states holding elections yesterday, Alabama Democratic Lt. Gov. Bill Baxley, a colorful populist seeking to succeed Gov. George C. Wallace, and state Attorney General Charles Graddick, a former Republican, won the two spots in a June 24 runoff.

With 70 percent of the precincts reporting, Baxley had 36 percent, Graddick had 30 percent, former governor Fob James had 21 percent and former lieutenant governor George McMillan had 12 percent. Since neither of the top two candidates won 50 percent, they are forced into a runoff.

In the Democratic Senate primary, conservative Rep. Richard C. Shelby avoided a runoff by defeating Jim Allen Jr., son of the late senator, by a 52 percent to 35 percent margin.

In Iowa, former state Senate majority leader Lowell Junkins easily won the Democratic gubernatorial primary with 52 percent over three others and will oppose Republican incumbent Gov. Terry E. Branstad. In the Iowa Democratic Senate primary, John Roehrick, a Des Moines lawyer, took a big early lead over Juan Cortez, a follower of Lyndon H. LaRouche Jr.

Democrats hope to make the race against Branstad, who is vulnerable because of the farm economy, a test of President Reagan's farm policies. Republican incumbent Sen. Charles E. Grassley was unopposed for renomination.

In the race for the House seat of Rep. Berkley W. Bedell (D-Iowa), who is retiring, actor Fred Grandy, who played "Gopher," the bumbling purser on the television program "Love Boat," easily defeated two opponents in the Republican primary. Bedell's aide, Clayton Hodgson, defeated four opponents for the Democratic nomination.

In New Mexico, Ray Powell, a retired vice president of Sandia National Laboratories, was declared the winner of the Democratic gubernatorial nomination over two write-in opponents. In the Republican gubernatorial primary, Garrey E. Carruthers, former assistant interior secretary in the Reagan administration, defeated five other candidates. With 60 percent of the vote in, Carruthers had 34 percent, former state representative Colin McMillan had 27 percent, former state senator Joseph Mercer, 19 percent, former state representative Frank M. Bond, 12 percent, former state senator Paul Becht, 7 percent, and William Jay Loomis Jr. had 1 percent.

Rodino, chairman of the House Judiciary Committee, is expected to have no trouble defeating Republican Alvin K. Terry for his 20th term.

"Thank you, thank you very much," Rodino said last night. "The key for Peter Rodino is that for 38 years he has kept his commitment to serve the people, all the people, regardless of race, color or creed. I knew that the people were going to put aside any thought of race. They rejected it."

He had argued that with his experience and influence, he could do more to help black constituents than any freshman congressman could.

"This is a resounding victory for the people," he said.

Payne was winning only in his home ward despite heavy campaigning on his behalf by the Rev. Jesse L. Jackson.

"In the history of New Jersey, there has never been one black congressman," Jackson complained. "That's not fair."

California

An election-eve California Poll showed Herschensohn and Zschau running even statewide with 11 other candidates trailing far behind in the hard-fought struggle for the Republican Senate nomination. Cranston was certain of renomination. The major candidates for governor, Democratic Mayor Tom Bradley of Los Angeles and Gov. George Deukmejian, the Republican incumbent, had only token opposition.

In the poll, Zschau and Herschensohn had 26 percent each but Herschensohn had a 45 percent to 19 percent lead over Zschau in Los Angeles and Orange counties, the state's largest population center. Zschau, a former electronics entrepreneur, led Herschensohn by 42 percent to 3 percent on his home turf, the Bay Area.

The other top statewide issue was over Proposition 51, known as the "Deep Pockets" initiative, which would limit liability claims in multidefendant lawsuits. Trial lawyers and consumers opposed it; insurance companies, doctors and local governments were for it. The returns show voters backing it by a wide margin in 56 of 58 counties reporting.

Other candidates in the Republican Senate race included Rep. Bobbi Fiedler, former Los Angeles police chief Ed Davis, Los Angeles County Supervisor Mike Antonovich, supply-side economist Arthur Laffer and former Black Panther Eldridge Cleaver. Fiedler and Davis were among the early leaders but both were badly hurt when Davis accused Fiedler of trying to bribe him to quit the race.

South Dakota

Abdnor's contest with Janklow, a colorful and aggressive politician, was surprisingly low-key but grew heated at the end. Janklow argued that Washington -- including Abdnor -- was primarily to blame for the state's economic woes but contended that he was Abdnor's friend and didn't want to hurt or embarrass him. Abdnor stressed his hard work and ability to get things done for South Dakota because of his years in Congress. He said Janklow's reputation as a harsh critic of Congress and federal policies would make him an ineffective outsider.

Rep. Thomas A. Daschle, the Democratic candidate for the Senate, was unopposed for nomination and is regarded as a formidable foe because of his organizational skills and his reputation for being out front in support of farmers.

One issue that came up in the last 10 days was the decision of the national Realtors' political action committee to spend $100,000 on Abdnor's behalf. Janklow charged that the move was engineered by Democrats who believed Abdnor would be an easier opponent for Daschle in November but the real estate agents said their support was due to Abdnor's seats on the Joint Economic Committee and the Environment and Public Works Committee.

In the Democratic gubernatorial race, Kneip had an 18-point lead in one statewide poll and had been widely predicted to win. But some in the state warned that he was acting as though he had already won the nomination. His opponents also accused him of running out on the state when he resigned as governor to accept an ambassadorial appointment in the Carter administration.

Alabama

The battle to succeed Wallace, who has dominated state politics for a generation, will continue in a June 24 runoff, with Lt. Gov. Baxley, running as a liberal populist, in one slot and Attorney General Graddick, a conservative, in the other.

Graddick, running as a "fresh-face" candidate, withstood a late challenge from former governor James, who closed his campaign with a colorful whistle-stop train tour of the state.

Former lieutenant governor McMillan, who narrowly lost to Wallace in the 1982 Democratic runoff, was never able to ignite his campaign this year and trailed far behind the leaders.

Baxley, 44, an ally of Wallace in recent years, ran with the endorsement of the state's two major black political organizations, the teacher lobby and organized labor.

A runoff with Graddick, the only Democratic candidate not to court black votes, poses a clear liberal-conservative fight with racial overtones.

The 66-year-old Wallace, paraplegic since a 1972 assassination attempt, announced tearfully in April that he had "climbed my last political mountain" and would not seek an unprecedented fifth term.

In the Democratic primary for the Senate nomination, Shelby, a fourth-term conservative from Tuscaloosa, avoided a runoff with Allen, son of the late senator, and will challenge freshman GOP Sen. Jeremiah Denton. With 70 percent of the vote counted, Shelby had 53 percent and Allen 32 percent.

The campaign drew to a sizzling close when Shelby, who spent $1 million on his campaign, branded Allen, who had drawn dangerously close in polls, "an admitted drunken driver." Allen, 40, a state board of education member from Gadsden, cried foul, saying he pleaded guilty to the drunk-driving charge while in college and no longer drinks.

Denton, the first Republican senator from Alabama in a century, had only token opposition.

In the Republican gubernatorial primary, Guy Hunt, a former probate judge, easily defeated Birmingham businessman Doug Carter, a former Wallace speechwriter. No Republican has held the Alabama governorship since Reconstruction.

New Jersey

With no Senate or gubernatorial races, interest focused on the primary challenge to Rodino, whose district now is primarily black. Black leaders had argued that it was time for the district to be represented by a black, despite Rodino's record of support for civil rights. The organization that earlier helped defeat veteran Newark Mayor Kenneth Gibson also supported Payne.

Rodino campaigned heavily in the final three weeks, amid warnings that the race might be close. He spent nearly $30,000 on campaign commercials. Prominent blacks including two congressional colleagues, Reps. Charles B. Rangel of New York and Mickey Leland of Texas, came to his district to endorse him and praise his leadership on civil rights issues.

Payne, by contrast, raised and spent little. His argument to the voters was based primarily on race -- that it was time that a district that had more than 60 per cent black population be represented by a black. New Jersey has never sent a black person to Congress.

Iowa

Former state Senate majority leader Junkins, who pledged to make the general election a referendum on the state's ailing farm economy, easily won the Democratic gubernatorial nomination and the right to challenge Branstad.

Junkins, 42, a farmer and ambulance business owner, made a $1.5 billion economic development plan a major issue in his race against Lt. Gov. Bob Anderson, 41, a former high school journalism teacher, and state Sen. George Kinley, who owns a Des Moines golf shop. With almost half the vote counted, Junkins had 54 percent, Anderson had 34 per cent and Kinley had 11 percent.

Branstad, Reagan's 1984 state campaign manager, has sharply criticized the president's farm policies, but is considered far more vulnerable than Grassley, the state's most popular Republican officeholder. Roehrich, who easily defeated LaRouche follower Cortez, a 69-year-old labor leader, for the Democratic Senate nomination, is considered a longshot against the GOP incumbent.

In congressional races, actor Grandy sailed to an easy victory in a GOP primary for the seat held by retiring Rep. Bedell, a Democrat. Four candidates sought the Democratic nomination in a fight that was undecided in the early morning hours.

In the race for the state's other open congressional seat, former state Democratic chairman David Nagle defeated state House Majority Leader Lowell Norland for the party's nomination for the spot held by retiring Rep. Cooper Evans, a Republican. State Rep. John McIntee was unopposed in the GOP primary.

Mississippi

Reuben Anderson, the state's first black Supreme Court justice, beat back a challenge by Richard Barrett, a white segregationist. Mike Espy, a black lawyer, held a wide lead over two white opponents with well-known political names in a Democratic congressional primary in the Delta.

With three-fourths of the vote tallied, Espy had 52 percent; Hiram Eastland, cousin of late Sen. James O. Eastland, had 25 percent; and Pete Johnson, whose grandfather and uncle both served as governor, had about 24 percent. Espy, the first black appointed to the state attorney general's staff, can avoid a runoff if he receives more than 50 percent of the vote. GOP Rep. Webb Franklin, who faced no primary opposition, now represents the black majority 2nd District.

North Carolina

Lester D. Roark, former deputy state attorney general and ex-mayor of Shelby, won 65 percent of the vote to defeat Belmont Mayor Jack L. Rhyne in the Democratic runoff for the 10th District congressional seat held by Rep. James T. Broyhill, who is running for Senate. Roark will face state Sen. Cass Ballenger, a Hickory businessman.

New Mexico

Carruthers defeated a crowded field in a bid to become the state's first Republican governor in almost two decades.

Powell, a businessman in his first run for office, won the Democratic nomination over two write-in opponents. Democratic Gov. Toney Anaya, whose adminstration has been plagued by corruption and a sagging economy, is prohibited by law from seeking a second term.

Montana

There are no Senate or gubernatorial races and no primary challenges to the two representatives, Democrat Patrick Williams and Republican Ron Marlenee. Madison County Sheriff Johnny France, famed for his single-handed capture of two self-styled "mountain men" who kidnaped a Bozeman woman, was defeated in the Republican primary by one of his deputies, Dick Noorlander. Noorlander, who accused France of not spending enough time on the job, won by 1,120 to 919, according to unofficial returns.