Da Gelsomina is the best restaurant in Capri. Off the beaten path but worth the trip. Go to Anacapri and the restaurant will come and pick you up. Food is fresh and local. Home made pastas and fresh seafood. Brick chicken is one of their best. Grilled mozzarella as a starter is nice and flavorful. Try the grilled vegetables as well. You'll want to go back.
Villa Verde is a nice casual place with traditional Italian dishes - relaxed and not to formal with good food and an extensive wine list. The ribeye steak had great flavor.
Aurora is a top restaurant -more formal. The pizza is great if you like it very thin. Stuffed zucchini flowers were amazing. Rice stuffed torte was tremendous as well.