Last year my boys were able to participate in a wonderful FREE program offered by the YMCA of Grearter New York called Splash Week. This year Splash week is from March 29 - April 2. Each Y has varying times and hours for the classes so call to find out what is still available and what will work for you. I know the West side YMCA at 5 West 63rd street, where we attended lessons last year was a wonderful facility and a wonderful experience. There is a hotline for Splash Week registration (212) 875-4214 at thw west side Y. I made a few calls to other Ys and the Jamaica Y's program is full and others have limited space or limited classes available.
My boys were registered for Free swim lessons offered by the Department of Parks and Recreation for the Spring session. Unfortunately the pool they were attending- Roy Wilkins Pool - in Queens- was flooded and now they have no idea when it will be available again for lessons. The last time the pool was flooded it took several years before the pool was opened again. In fact they actually received a new pool. In the summer we take advantage of the free swim lessons and have explored a few of the pools in Queens.
There is also a homeschool group that has lessons at the Asphalt Green in Manhattan. These classes are conducted by excellent teachers. I have experience with the Asphalt Green from my Brearley days both going to school and working there in the athletics department.
There are other onderful swim classes and instructors and facilities out there in all boroughs and long island. Feel free to chime in if you have any advice.