South side of Capri town: Villa Krupp with park drum around, high above the cliffs overlooking the scenic rocks. From there, a trail about 5 km in the cliffs along the east side of the island goes back to the city/harbor. Very impressive.
Anacapri (High Capri) for hikers: Along the harbor to the west, main road a bit high up to the stairs. At the end of the stairs turn right onto the main road until the next stairs. This goes up to Anacapri at the Axel-Munthe Museum with breathtaking sea panoramas. In Anacapri, after 250 m turn left to Monte Solara, signposted. You can also take the cable car and be reward with a unique panoramic view.