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.

Hospitals
Urgent Care/Walk-In Clinics
Walk-In/Emergency Dental Clinics
Low Cost Options
More Information on Affordable Healthcare

Hospitals
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

Urgent Care/Walk-In Clinics
http://citymd.reachlocal.net/
http://www.medriteurgentcare.com
http://walk-in-clinic-ny.com/walk_in_clinic.html
http://urgentcareny.com/site6/

Walk-In/Emergency Dental Clinics
http://www.centralparkwestdentistry.net
http://www.212smiling.com/emergency-care.php
http://www.emergencydentistnyc.com/

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
http://www.ahirc.org/newyork
http://www.nyc.gov/html/hia/html/resources/services.shtml
http://www.villagevoice.com/2003-12-30/nyc-life/free-and-low-cost-health-care/