A few East Hampton recommendations, based on my summer there last year:
I love Babette's for lunch or dinner, casual and friendly place, good selection of healthy food, but not so healthy you don't want to eat it...fish, grass fed burgers, vegetarian dishes etc. It's overpriced, but so is everything there.
Also there but more appropriate for dinner (may not even serve lunch), the restaurant Living Room at the inn c/o The Maidstone is just fantastic. It's Swedish (or Swedish influenced) on account of the hotel's owner. Ate there several times and stayed at the hotel a couple years ago for Labor Day - I just love the place. Eclectic and trendy, yet comfortable and homey-feeling at the same time.
For beaches my favorite is Wiborg, I found it to be the emptiest of anywhere in EH, and if you need a snack or a bathroom you can walk over to Main Beach pretty easily (though be prepared to drool as you drive past the Maidstone Club).