I have not done the trip that way, but I used to sail the South Shore extensively. Because I was homeported there, I didn't stay at marinas either. But the bays inside the barrier islands have plenty of good anchorages. The "State Boat Channel" is Long Island's section of the ICW. It runs through numerous islands and salt marshes that provide good sheltered anchorages. The water is not deep, the bottom is generally sand or mud, sometimes grass.
Marinas in NY are expensive, but there are plenty of great towns on the bay to visit. On the Fire Island side there are several town marinas and beach communities, bring mosquitoe repellent.