The suggestion about the gallery guide is excellent. You can purchase a copy in most galleries, and it makes a nice (and lightweight) souvenir to take home. One of the features of the book is that it includes street maps that locate the galleries and also identifies subway stations. But I suggest spending time at the web site looking at the listings, and preparing a list of the galleries you want to visit, before the trip.
The gallery cluster is west of 10 Avenue, and between 19th and 25th street. You can easily spend a full day wandering around that area. There are also a few galleries uptown - some on 57th street.
There are a lot of great restaurants in New York. When we werre last there in December, we wandered into the Chelsea Square Restaurant at 23rd and 9th. It reminded me a lot of what we call a diner in our area - extensive menu, enormous portions, and relatively inexpensive. I ordered flounder and I almost couldn't finish the meal.
Another place we have enjoyed is Mustang Harry's on 7th Ave, around 30th street. It's easily within walking distance of the gallery district and on the direct route from Penn Station.
David mentioned the John Stevenson Gallery - between 8th and 9th on 23rd. That's a must-see on our visits. He specializes in high quality (high-priced) photography, mostly platinum and other alternative processes. And it's a nice (air conditioned) place to stop and coiol off while walking from the subway.
There is a smaller cluster of galleries in SoHo, mainly along Broadway. In particular, the building at 560 Broadway has a number of interesting galleries, and just a block north at 670 Broadway is another building that includes the Leica Gallery. There are several good restaurants in this area - Zoe and Milady's (both on Prince Street west of Broadway) are examples we have enjoyed.