Corsica is famous for its superb beaches, so allow at least a few days in your journey for doing nothing. Explore Tamarone beach, one of the most beautiful on the island. The Vasques de la Solenzara have some beautiful beaches next to natural swimming pools. It's best from May to September.
If you like thrills, Corsica is made for you. You can go climbing or on the via ferrata of the Asco valley, or explore the Corsican countryside on horseback. There's also whitewater kayaking in Aléria or the Bavella canyons. Its best between April and October.
Corsica is the ideal destination for divers. It's best between April and August when the air temperature is pleasant and the same goes for the water. You can chose from the Gulf of Ajaccio or Porto, both of which are famous for their diving. At Bastia you can dive among the wrecks.
Some of Corsica's little villages are worth a trip by themselves. Pigna and Sant’Antonino, for example, are among the "most beautiful villages in France" and Piana is well-known for its calanques. And, of course, you shouldn't miss Sainte-Lucie-de-Tallano, one of the island's most beautiful villages, at 450 m above sea level. It's best to going between April and October.
If there is one activitiy Corsica is particularly known for, it's walking. Walk the 15 days of the legendary GR20, or simply walk one of its paths. Explore the calanques of Piana and Capo Rosso or go along the cliffs of Bonifacio. It's best in May, June or September.