As far as grocery stores, the store next to the Cool Spot restaurant has a great selection of spirits, wine, beer, and has cereal (hot or cold), milk (boxed or refrigerated) and yogurt, snacks. I always frequent that store when I'm passing by, it has a lot of good selections. Shamrock is another good store to visit, but that's further into town.
My favorite restaurant in that area is Shereta’s Restaurant located at the Sea Wind. Excellent food, great views ocean side. You mentioned your staying at the Palms, make sure you ask for a refrigerator, seeing you may be getting some perishables, plus its nice to have some things cold to drink. Have a pleasant reach!.