7:37 pm, today, 3/29
"March 30, all New Yorkers age 30 or older will be able to be vaccinated, and all New Yorkers age 16 or older will be eligible on April 6"
Having been through the system earlier, don't wait to be called by your primary physician or use drugstore. Make an appt the first day eligible though the NYS site for a large NYS facility. Don't pick a day with just a few openings- they'll be gone by the time you finish the application. Pick a day that has hundreds of available appointments.
As of last week you can now get a CDC phone app that will give you a certificate of vaccination with a QR code that can be read by event venues, etc after you've been fully vaccinated.