I've never been to Block Island though I've considered stopping there on my way to Cape Cod. Like someone else said, I think it would be nice for a few days, but any longer and it could get boring.
I love Cape Cod, unfortunately you'd be getting there right at the beginning of their busiest time of the year. June and early July are the best times to be their in the summer. August is their busiest time.
For a quiet, private, residential area I stay
HERE. It's owned by an older couple who are really nice and helpful. They used to have a downstairs apartment with a private entrance and patio and nice backyard. I don't see it listed on this site though. They have a private beach that you have to trek down about 3 flights on stairs to get to, but you'll probably be the only people there. I have a pic I took just like the one on the site.
I like to drive from there to Woods Hole and take the bike by ferry to Martha's Vineyard and just spend the day biking all over the island taking pics, shopping, and eating homemade ice cream. A lot of people like to rollerblade around there too. You could also rent a moped, but I don't really care for them. You can take the ferry from there to Nantucket as well, but it's a long trip. It's shorter to Nantucket from Hyannisport, but I don't like Hyannis at all. It's about the only place on the Cape that has any kind of crime rate.
The people on Cape Cod are generally very friendly and laid-back. Lots of clambakes on the beaches at night. You should take your bike up to the Province Lands and try riding through those dunes. I hope you're in good shape. In fact, Cape Cod and both the islands are great for bike riders. Lots and lots of beaches on Cape Cod and MV and Nantucket to pick from, some crowded and some not, great for pics. Whale watching and puffin watching are nice depending on how successful the tour is.
Falmouth is a great place to stay if you want to be in a little larger, but laid back, town and have quick easy access to the ferries for MV and Nantucket. The Seaside Inn there is a nice place to stay considering the price. The Red Horse Inn is nice too, but it's a motel, not an inn and it's a little more expensive.
Sorry I can't help you with Block Island, but if you go to Cape Cod you won't run out of things to do, that's for sure. Hope you have a good time, wherever you go.
Edit: P.S. The only places I would recommend you not stay in are Hyannis and Provincetown.