Try Apizz. My husband and i went last week. The white pizza was gorgeous - heavy ont he parmagiano, then we had the whole fish (sea bass) which was stuffed with lemons, rosemary and thyme. It was very good. Nice wines. adorable and romantic spot on the lower east side. excellent Italian.. It feels special.
go to max's and sit outside in the garden. it's italian and about $12 for a plate of pasta that your grandma might have made you. really excellent, not fancy but young and vibrant. and they have a nice selection of italian wines. it's in the east village. then nearby, you can walk to cocoa bar for a quality version of max brenners if you are in the chocolate frame of mind. get the spicy brownie. you won't be sorry. for japanese tapas try kasadela on avenue C and 11th street (?) , for the best sushi you have had (27 in zagats) and a total bargain try kanoyama on 2nd ave @ 12th street. pylos is fabulous and moderately priced greek food on 7th street at ave A. so delish. For a great burger for only $10 includig a beer (!!!) go to the clinton street bakery. that is a fabulous burger and they even bake their own brioche rolls. enjoy new york. avoid midtown if possible - with one exception which is right by your hotel. there's a new tiny french cafe called macaroon. stop there for a croissant and a beautiful coffee before you head out on your day. it's on 36th street between 7th ave and broadway.
in addition to the tompkins square market mentioned below, there is a great italian grocery called russos on 11th street between 1st and 2nd ave. they have lovely mozzerella, sausages, fresh and frozen pastas of every variety. good olives. as for bakeries, you have to go to the city bakery oupost which is on 1st ave at 13th street to get something decent. that pinisi bakery on 4th street is terrible and whole foods bakery is really only average.
I agree with the riviera bake house in ardsley - particularly their almond croissants. in manhattan you can find amazing croissants at bergomont on 9th ave and 20th street and at falai oon clinton street