You will get an ear full positive and negative from the LICC guys about Champlins. But if you're looking for a Marina on BI and have kids, that's the one. Shops, games, pool, etcc.
As for MV, of the three main choices, of Vineyard Haven, Oakbluffs, and Edgartown, Oak Bluffs is your best bet. The town runds the Marina in the inner harbor. From there, all the attractions of restaurants, stores, carosel, etc.. are right there. All though be aware that pedestrians walk along the sidewalk and piers and can look right down into your cockpit and cabin. Not very private if that's what you're looking for. Vineyard Haven is a nice town with several Marinas. None of the names come to mind. I even stayed at one. The town is nice and fun, but not as kid oriented as Oak Bluffs. Edgartown is a great town. But not really kid oriented and they have very limited slip space, mostly for bigger (50ft+) Yachts. Most transient space is moorings.
Nantucket I believe (have not cruised there yet) the only option is Nantucket Boat Basin. All slips are right in the heart of the town. That's a quick rundown. Hope it helps. Chime in with other questions.