We just spent a week on Nantucket at the end of June. Totally agree "fabulous island with incredible food"...my favorite place on earth! Beaches are excellent, rent a bike and ride the paths, shopping is great too.
Personal experiences that are places I would highly recommend -
Black Eyed Susan's
Centre St Bistro (weekends only for breakfast)
Downyflake donuts (out near rotary)
Nantucket Bake Shop (Jamaican patties, coco bread, croissants)
Daily Bread (breakfast pastries)
PROVISIONS sandwiches, esp. the turkey terrific
Fog Island Cafe
Sconsett sandwich shop/stand (not cafe ot market)
Nantucket Lobster Trap (try their lobster and elbow bender rum drinks!)
Straight Wharf (our splurge anniversary meal - food, service and ambiance are top notch. also have great drinks and apps at the bar, can't wait to go back!).
We ate at the Starlight Cafe before a movie, theater was fun, food not so good, although the turkey burgers were fine.
The Juice Bar is the best place for ice cream, worth the wait and try the waffle cone.
We enjoyed afternoon drinks at Cambridge Street and would like to go back to try their apps/pizza. Less expensive, more casual place.
Even Keel Cafe
Food For Here and There (pizza)
Bring some money and a lot of it. Fabulous Island with incredible food. You must have reservations or you will never get in to eat. Try Black Eyed Susans. The Tuna Tartare is the best! Also try the pearl. I had dinner there last week and it was an outstanding experience. The food and beverages are perfection. Go to Aunt Leahs for the best homemade fudge. The Juice bar is always filled with people in line for ice cream. I do not know what I love more... The shopping or the food in nantucket. The shopping is killer! Do not forget your money and again bring a lot of it, it's worth it.