Capri can be split in two parts : an eastern (Capri) and a western (Anacapri). The latter is also the highest part of the island, reaching altitudes above 500 m. You only get there via a winding road, going high up via the northern side of the cliff. You can take a local bus for no price, or one of the numerous busses waiting for tourists in the port.
The centre of Anacapri is not large. Along the Via Orlandi, there a few worthwile places (Piazza la Torre, Chiesa San Nicola, Piazza Diaz), but the main attraction is certainly the Villa San Michele. This was the residence of the Swedish doctor Axel Munht, who constructed the villa on the remains of old chapel dedicated to San Michele.
Munthe was not only a medecin, but also an art lover, who collected many treasures from the Roman era (which he received from the local farmers when they dug up a strange stone while working on their fields). On top, he was a known ornithologist. Part of the island is protected as Natural bird reserve, a result of the actions of Axel Munthe to rescue the migrating birds from the local bird shooter on the island.
From Anacapri, you can make a few beautiful walks. A first one leads in about half an hour to the Monte Solaro (that you can reach in 10 minutes with the télépherique. With its 589 mètres, this is the highest point of the island.
Close to Monte Solaro is Cetrella. Fro Chiesa Santa Maria, you can enjoy one of the most beautiful views of the island. The start of this walk can be found on the way from the Piazza la Torre to the Villa San Michele.