So what are you looking to do?
Fast and Fancy and you need to stop in Newport.
Block is great idea. Be sure to go into New Harbor. June won't be crazy busy yet, but look into the Block Island Race Week. It can be busy.
If you're willing to continue further West, think of Mystic, CT.
Point Judith is okay. Lots of room to anchor outside in the refuge. There are a few marinas up the creek.
Or, head up Narragansett Bay to Jamestown. Great little town. Get a mooring at Conanicut Marina. They also have a ferry to Newport.
Alternatively, anchor in Dutch Harbor on the West side of the island and walk about one mile on one road over to Jamestown.
Wickford, RI is a very protected harbor with many docking and mooring options.
Bristol, RI is a great town too. Herreshoff museum is great, if you like that sort of thing. Aiden's pub has never disappointed anyone that I know.
If you want quiet and bbq on the rail, think about Potters Cove on the Northeast corner of Prudence Island in the Bay.
From Cutty, its pretty easy to get over the Martha Vineyard. Menemsha is easy. You many even get inside with the 37. Otherwise, outside it good with southerlies. Vineyard Haven is reachable. I prefer Edgartown, but further.
Probably have a thousand other ideas. Narrow me down a bit.