There are many supermarkets in every borough.  The best way to find a supermarket near your residence is to ask your neighbors or go to www.superpages.com, where you can get a list of supermarkets in your area.  There are also many supermarkets in Manhattan.  Union Square has a nice farmers market on Wednesdays and Saturdays!  For fresh and cheap vegetables, take the subway to Astoria, Queens at Broadway/30th Ave/Ditmars stations.  There are many nice outside markets at these stations.