Santa Cruz has a nice little museum—old farm equipment and other items demonstrating the island’s history.
During the summer months on Graciosa, street bullfights take place all over the island. What are they? Well, they are events where bulls (one at a time) run up and down a street chasing those that are courageous/stupid enough to get in it's path. It's a chance where everyman gets a chance at the "prime-time" glory! Please do not confuse this with the running of the bull in Pamplona. They are not the same thing. Usually lots of people come to these events. It is highly entertaining, although it does have it's dull moments. You do not have to worry about the bull getting hurt, it doesn't. On the other hand people do get hurt and sometimes killed. Use every ounce of common sense in you at these events. Oh by the way, bulls can jump! And they will! eh touro! eh touro!
You will find the Queijada factory in Sao Mateus (Praia). These pasteries are special!
What to buy: At the end of your trip, buy as many queijadas as your luggage will permit you. They come packaged in a box of 6. They will still be fresh to eat by the time you return home.
What to pay: Something like $5-10 will buy you a dozen of these things.
If you have access to a boat, cruise around the Ilheu de Baixa (small islet) and other rocks just southeast of Graciosa to look for seabirds and other sea life.
Ponta da Barca is a giant stone whale just offshore—no imagination is needed.
To find it, just head toward the Ponta da Barca lighthouse, which is only a few miles from Santa Cruz. The lighthouse itself is off-limits to visitors--The big dog growling outside made that very clear.
The highlight on this island seems to be going down into Furna do Enxofre, the crater of an extinct volcano. There are a bunch of rough steps going steeply downhill, which gets you to the 186 steps leading down into the crater. No handrails. There is a huge sulphur cavern at the bottom, and an underground lake.
This tip is second-hand, as I wasn't physically able to do it. The people who went were impressed and glad they went. No photos either--I should have sent my camera down with somebody!
Equipment: Shoes or boots with good soles. It can be slippery in spots.