I live in PA, but my favorite places to shop are Trader Joe's and Whole Foods. Trader Joe's dried fruits are very affordable and very good quality. I prefer Whole Foods for the produce.

Welcome to the car-driving club, Kim, and good luck finding parking! I'll start with Queens and work my way outwards:

Sapori D'Ischia in Woodside, Queens, has an excellent supply of cheeses, cured meats, pastas, and other specialty products from Italy.

Despana Brand Foods on Northern Boulevard in Jackson Heights has a small, but good selection of products from Spain.
http://www.thefoodsection.com/foodsection/2003/10/source_despana_.html

A car makes it much easier to transport the fine beers at Park Slope's Bierkraft and Cobble Hill's American Beer Distributing Company.

Patel Brothers on 74th Street in Jackson Heights is a giant Indian supermarket. They have the largest supply of packaged flatbreads I have ever seen.

If you are in Long Island, there's a supermarket-size Fairway.

If you are in New Jersey, National Wholesale Liquidators, strangely enough, operates the International Food Warehouse, which has a huge selection of ethnic foods. Here's an account: http://www.chowhound.com/dinner/dinner.html

Also in NJ, Wegman's (there's one on the way to Princeton) is the biggest supermarket I have ever seen. Nothing particularly unique about it, but it is a sight to see when Key Food and C-Town are the norm.

I live in San Fran area and we are so spoiled! But one thing we don't have is your high-quality Greek stores. We are Greek and adore some of the stores on Ditmars. There's one--If I remember--called Angelo's. So nice that it is not all chains yet in that area. There is a HUGE Greek grocery at 31st st near Astoria Bl, just off the elevated line. Olives are out of this world. Try the ones from Crete!

Strangely enough, I end up shopping at Costco a lot.