Gov. Kathy Hochul of New York took a crucial step toward winning a full term on Tuesday, easily fending off a pair of spirited primary challengers and cementing her status as the state’s top Democrat less than a year after she unexpectedly took office.The runaway victory by Ms. Hochul, the state’s first female governor, sets the stage for what could be a grueling general election contest against Representative Lee Zeldin, a conservative congressional ally of former President Donald J. Trump who beat out three fellow Republicans in a gritty race for his party’s nomination.Ms. Hochul enters the November contest with deep structural advantages: She has the power of the governor’s office and overflowing campaign accounts, her party enjoys a more than two-to-one registration advantage and Republicans have not won statewide in New York since Gov. George E. Pataki secured a third term in 2002.But with warning signs flashing red for Democrats nationally and New Yorkers in a dour mood over elevated crime and skyrocketing prices for housing, gas and a week’s groceries, both parties were preparing to run as if even deep blue New York could be in play this fall.The general election contest promises to have sweeping implications that ripple well beyond New York in the aftermath of two recent landmark Supreme Court decisions that ended the federal right to an abortion and curtailed New York’s ability to regulate firearms. The state has long been a safe haven for abortion and had one of the most restrictive laws regulating firearms, positions Mr. Zeldin and Mr. Giuliani oppose and could try to change if one of them wins.With that fight looming, Democratic primary voters on Tuesday chose Ms. Hochul, a middle-of-the-road incumbent who spent the campaign’s final weeks casting herself as a steady protector of the state’s liberal values — if not the firebrand or soaring orator who have found success in other races.“We cannot and will not let right-wing extremists set us backward on all the decades of progress we’ve made right here, whether it’s a Trump cheerleader running for the governor of the State of New York or Trump’s appointed justices on the Supreme Court,” Ms. Hochul told supporters at a victory party in TriBeCa in Manhattan.Standing, symbolically, under a glass ceiling, a jubilant Ms. Hochul added that she stood “on the shoulders of generations of women” in her effort to become the first to win the governorship.Ms. Hochul planned to quickly return to Albany, where she has called the Legislature back for a rare special session to respond to the Supreme Court ruling invalidating a century-old state gun control law.The race was called by The Associated Press 25 minutes after the polls closed in New York.Ms. Hochul had won 67 percent of the Democratic primary vote, with 50 percent of the expected vote counted. Jumaane D. Williams, the left-leaning New York City public advocate, had won 21 percent of the vote. Representative Thomas R. Suozzi, a Long Island moderate who ran an aggressively adversarial campaign focused on cutting crime and taxes, won 12 percent of the vote.A Guide to New York’s 2022 Primary ElectionsAs prominent Democratic officials seek to defend their records, Republicans see opportunities to make inroads in general election races.Governor’s Race: Gov. Kathy Hochul is trying to fend off energetic challenges from two fellow Democrats, while the four-way G.O.P. contest has been playing in part like a referendum on Donald J. Trump.Where the Candidates Stand: Ahead of the primaries for governor on June 28, our political reporters questioned the seven candidates on crime, taxes, abortion and more.Maloney vs. Nadler: New congressional lines have put the two stalwart Manhattan Democrats — including New York City’s last remaining Jewish congressman — on a collision course in the Aug. 23 primary.15 Democrats, 1 Seat: A newly redrawn House district in New York City may be one of the largest and most freewheeling primaries in the nation.Offensive Remarks: Carl P. Paladino, a Republican running for a House seat in Western New York, recently drew backlash for praising Adolf Hitler in an interview dating back to 2021.Democratic voters also rewarded Ms. Hochul’s handpicked lieutenant governor and running mate, Antonio Delgado, who survived a spirited challenge from Ana María Archila, a progressive activist aligned with Mr. Williams. Mr. Suozzi’s running mate, Diana Reyna, was also on track to finish third.Mr. Delgado, a former Hudson Valley congressman, was only sworn in a month ago after the governor’s first lieutenant, Brian A. Benjamin, resigned in the face of federal bribery charges and after Ms. Hochul pushed for a legal change to get him on the ballot.His victory was a the night’s second significant disappointment for progressives, who saw Ms. Archila as their best shot at winning statewide office this year. Ultimately, she could not overcome the vast financial and institutional advantages that helped Mr. Delgado blanket TVs and radios in advertising.Primaries in other statewide races — for U.S. Senate, state attorney general, comptroller and the Republican nominee for lieutenant governor — were uncontested.Turnout was relatively low across the state, especially compared with 2018. Combined with President Biden’s slumping approval ratings, Ms. Hochul’s relative newness to office and strong Republican performances last fall in Virginia, New Jersey and on Long Island, the figures were enough to give Democrats cause for concern as they pivoted toward a general election.“Democrats better not take this for granted because Lee Zeldin is a wolf in sheep’s clothing,” said Isaac Goldberg, a New York Democratic strategist not working on the race. “He will appeal well to his fellow suburbanites who don’t know how far right he truly is.”Mr. Zeldin, 42, defeated Andrew Giuliani, who had captured far-right support based on his connections to Mr. Trump, his former boss, and the former New York City mayor Rudolph W. Giuliani, his father. Mr. Zeldin had 42 percent of the vote, with 55 percent of expected votes reported.He also beat Harry Wilson, a corporate turnaround specialist who burned more than $10 million of his own money into his campaign, and Rob Astorino, the party’s 2014 nominee for governor.The victory was a triumph for the state’s Republican establishment, which threw money and support behind Mr. Zeldin early — a wager that a young Army veteran with a track record of winning tight races on eastern Long Island could appeal to the independents and disaffected Democrats that Republicans need to sway in New York to have a path to victory.Mr. Zeldin has tried to orient his campaign around bipartisan fears about public safety and inflation, promising to open up the state’s Southern Tier to fracking natural gas, reverse the state’s cashless bail law and end coronavirus vaccine requirements, while accusing Ms. Hochul of doing too little to restore public safety.BDemocrats have already started amplifying Mr. Zeldin’s more conservative positions on guns (Mr. Zeldin once said he opposed New York’s red-flag law), abortion rights (he celebrated last week’s Supreme Court decision overturning Roe v. Wade), and, above all, his embrace of Mr. Trump and vote on Jan. 6, 2021, to overturn the results of the presidential election in key swing states.No Republican candidate who opposes abortion rights has won New York’s top office in the half-century since the state legalized abortion.For Ms. Hochul, 63, Tuesday’s vote was the first major test of electoral strength since she unexpectedly came to power last August, when Mr. Cuomo resigned as governor in the face of sexual harassment allegations.A Buffalo native in a party dominated by New York City Democrats, Ms. Hochul had spent much of career toiling in relative obscurity, briefly as a congresswoman from western New York and for nearly six years as Mr. Cuomo’s lieutenant governor.She moved quickly to establish herself as a political force as much as a governing one, leaving little doubt that she was the Democratic front-runner. She won the endorsement of nearly every major Democrat and labor union, assembled a $34 million war chest to vastly outspend her opponents on TV and glossy mailers and took pains to balance the concerns of Black and progressive lawmakers and New Yorkers fearful of crime when pushing for a set of modest changes to the state’s bail laws this spring.She had to withstand aggressive critiques from Mr. Suozzi on her right and Mr. Williams on her left, who argued that she was doing too little to address soaring housing prices or crime and portrayed the governor as another creature of Albany’s corrupt establishment.Polls also showed that Ms. Hochul’s decision to spend hundreds of millions of taxpayer dollars on a new stadium for the Buffalo Bills was especially unpopular with voters. Like fears about public safety, the deal could re-emerge as a campaign issue this fall.But in the primary contest, at least, it did not matter. Ms. Hochul’s winning margin and coalition closely resembled the ones that sent Mr. Cuomo to Albany for three terms: a strong showing in the New York City suburbs; upstate strongholds in Albany, Buffalo and Rochester; and among Black and Latino voters in New York City.There were also signs that her emphasis on abortion and guns was resonating with voters she will need to turn out in November.“We need someone who can stand up for women’s rights and safety in our schools and a cleaner environment,” said Rebecca Thomas, a financial consultant who cast her vote Tuesday morning in Manhattan’s Battery Park City, at the same site where Mr. Giuliani cast his ballot.Of her fellow voter, she added: “Wrong person, wrong time.”Téa Kvetenadze More