The Paperwork To Open A Business In New York City, In Order
Formation, federal registration, state tax registration, then the permits specific to what you actually do. Out of order, you wait.

The sequence matters more than the individual steps. Several New York registrations require a number issued by an earlier step, so doing them out of order means waiting on something you could already have had.
1. Choose and form the entity
Corporation, LLC, partnership or sole proprietorship. This is filed with the New York Department of State, except for a sole proprietorship trading under the owner's own name.
If you are trading under a name that is not the legal entity name, you also need an assumed name filing — the 'doing business as'.
New York additionally imposes a publication requirement on LLCs, which involves publishing notice in two designated newspapers in the county of formation. It is an unusual requirement, it costs real money in Manhattan, and it catches out founders who formed elsewhere and expected New York to work the same way.
2. Get the federal EIN
Free, from the IRS, usually issued immediately online. Almost everything downstream asks for it.
3. Register with the state for taxes
If you sell taxable goods or services you need a Certificate of Authority to collect sales tax, and you must have it before you make a taxable sale. Employers register for withholding at the same stage.
4. Then, and only then, the industry permits
This is where the timeline actually lives. A consultancy may need nothing further. A restaurant needs food service permits, a license for any alcohol, sidewalk café approval if relevant, and inspections. Trades need licensing through the Department of Consumer and Worker Protection.
The city's own business wizard will generate the specific permit list for your activity and location, which is more reliable than working from a generic checklist.
The thing that delays people
Almost every avoidable delay comes from starting the permit stage before the entity and tax registrations are complete, because the permit application asks for numbers you do not yet have. Do the four stages in order and the process is administrative rather than adversarial.
Sources
NYS Department of State — Corporations