Generally speaking, you need to collect a minimum number of signatures on government-approved petitions to qualify for the ballot in New York.
If it were only that simple...
The State of New York has complex electoral laws which vary by office, party, county and city.
Through personal involvement in many NY political campaigns over the past 25 years, I can tell you that getting on the ballot in NY is no easy task for outsiders (i.e., candidates not backed by a party).
The petitions come with very specific requirements for use, such as the times they can be circulated and the definition of a valid signature (read more). A lawyer with experience in Suffolk County (CD-1) election law is highly recommended. That will cost you money.
Then, because you're not simply running for an open seat, but to replace a sitting congressman, you can expect the "establishment" to come down hard on you. They will do everything they can to protect "their man" (in this case, Congressman Lee Zeldin).
If they perceive you as a threat, one big thing they typically do is attempt to throw you off the ballot. They do this by submitting a legal challenge to your petitions on the grounds that they contain fraudulent or otherwise invalid signatures. Even one bad signature can get the entire petition sheet thrown out by a judge. Part of the strategy is to force you to defend your petitions, which can be costly and, therefore, destructive to your campaign.
All that said, the best way for you to find out exactly what it takes to get on the primary ballot is to: (1) hire a good election law attorney, or (2) contact the Suffolk County Board of Elections and ask them to provide you with the rules for ballot access, candidate qualifications (such as residency requirements in the district), deadline dates, and filing fees.
Then contact the Suffolk County GOP and ask them for a copy of their by-laws and candidacy requirements.
Consider using a fake name when you call these organizations. They all know each other. In NY, Board of Elections personnel are often appointed by party and political leaders. The BOE is a haven for patronage jobs (reference).
Or just use your real name, and let the fun begin. You will become famous very quickly among your opposition. The Congressman himself will probably know your name within the hour. Good luck!