Unless you're here for the diving, you'll only need to allow a few hours to "do" Giannutri -- that is, hike the island end to end, go for a swim, and maybe have a bite or drink at one of the island's few restaurants.
Giannutri has two boat landing sites, Cala dello Spalmatoio in the northeast (along the inner curve of the half-moon) and Cala Maestra, where the ferries land; this is a naturally protected notch along the northwest, or outer curve of Giannutri. These harbors are also where the only swimming beaches are; they're pebbly, but the water is splendid. A short walk from Cala Maestra puts you at the ruins of the Roman villa (always open; free) that was built on the panoramic terrace, surrounded by Aleppo pines, 1,900 years ago by the descendants of Nero, the Domitius Ahenobarbus family. The most salient feature of the site are several marble columns with Corinthian capitals standing around an impluvium (basin for collecting rainwater); fragments of black-and-white mosaics represent marine deities and creatures, and the extensive masonry of the villa's substructure and canals for water distribution still remain.
Cala dello Spalmatoio is where most of the "action" is on Giannutri, although of course that's a relative term -- but this is where you'll find a bar serving coffee, snacks, and alcohol; a restaurant/pizzeria; and a general store.
Many grottoes dot the island's rocky coastline, which is accessible only by sea, so inquire about informal boat tours at the port when you arrive. Unlike the other larger Tuscan islands, you can't easily rent boats on Giannutri for self-piloted exploration. The most famous area for grottoes is the aptly named Cala dei Grottoni (Cove of the Big Grottoes) along the curtainlike rock wall of Giannutri's southernmost tip.
Note: This information was accurate when it was published, but can change without notice. Please be sure to confirm all rates and details directly with the companies in question before planning your trip.