Financial district grocery
Also check out Barbarini Alimentari on Front St, near the Seaport. Great gourmet Italian grocery, stocks lots of pastas and other specialty items imported from Italy. There's also a fairly robust selection of cheeses from Artisanal.
Found it this past weekend while wandering, the man at the cheese counter was extremely friendly and gave me several unsolicited samples.
Zeytuna's is my favorite market in the neighborhood. The produce is fresh and reasonably priced and the butcher knowledgeable and friendly. Associated is good for stapes and non-perishables. Jubilee is the local 24-hour market for those last minute goods you might need for a midnight snack. Both Jubilee and Zeytuna's also deliver as necessary.
I've purchased meat at Jubilee before, but never anything complicated. I remember the meat as overpriced for the quality. At Zeytuna's, however, I've traded recipes with the butcher, gotten the butcher to clean up chops and gotten recommendations. When they first opened I actually looked forward to my chats with the young French butcher who would set me on a new cooking quest each week.
Associated and CTown are the best and least expensive for staples. Associated is in Southbridge Towers and CTown is just past the Brooklyn Bridge. Associated delivers if you are not a walker. Not sure about CTown. Jubilee is expensive. Zeytuna and Chinatown (very inexpensive, good for fresh fish too if you will be cooking while you are here) generally have the best produce.
pretty much every grocery store delivers. i know jubilee on john st does. fresh direct is good for everything except for anything fragile, like eggs. the associated on fulton is relatively inexpensive but kind of ghetto and you want to be careful about the freshness of their breads and produce. zeytuna is good for produce and meats but their packaged goods are overpriced. another good place is the amish market in tribeca.
You might want to do a search of this Board, this has been discussed. To get you started:
Whole Foods in TriBeCa
Associated, Jubilee, Zeytuna all in FiDi
CTown near Chinatown
Chinatown (short walk)
Financier (3 locations downtown for French pastries)
Union Square Greenmarket and Trader Joes are just one subway stop away
Fairway in Red Hook Brooklyn (not sure what the current ferry situation is)