Greenport has several options - Mitchell which is kind of exposed despite a sea wall but right in the middle of town and then the protected Stirling Basin which as Brewers, Stirling Harbor Marina, owned by Brewers and Townsend Hall Marina. There are also transient moorings and a town landing for your dink. To the East you can anchor
in Orient Harbor
Across is Dering Harbor, transient moorings, and Dering Harbor Marina. Not much going on shoreside, but the Island is interesting to tour from bikes (rental available)
On Shelter Island you have Coecles Harbor Marina, moorings and anchorage.
Down on the south side you have the anchorage at Smith Cove, and the more protected West Neck Cove and then there's Sag Harbor, transient moorings, marinas
outside the breakwater.
To the east a bit is the protected 3 Mile Harbor, anchoring
but not much going on shoreside.
If you are going to hit the spots, try working your way around Shelter Island on one side to your western most destination
and then sailing back the other side of SI hitting the other spots. There's plenty to see ashore and very pleasant sailing out there. At the pinch points you'll likely experience some large wakes from stink pots in a hurry.