Although I definitely recommend finding your own primary care provider that you know, love, and trust, when moving to a new city, that might not be top on your list of priorities. In the mean time, you can find information on available healthcare options in NYC.
A list of all New York City hospitals by borough: http://en.wikipedia.org/wiki/List_of_hospitals_in_New_York_City
Public hospitals in NYC are run by HHC, find info about them here: http://www.nyc.gov/html/hhc/html/home/home.shtml
The best hospitals in NYC and surrounding areas, according to US News & World Report: http://health.usnews.com/best-hospitals/area/new-york-ny
Low Cost Options
New York University’s Free Clinic: http://nycfreeclinic.med.nyu.edu/
Low-cost Dental Clinics: http://www.nyc.gov/html/hia/html/resources/services_dental.shtml
More Information on Affordable Healthcare/Coverage