If you have a week to devote to this region, try to visit two islands. Start by exploring Mytilini, the capital of Lesvos, a bustling center of commerce and learning, with its grand old mansions overlooking the harbor. From there, head to the countryside to the northern destinations: Molyvos, a medieval town sprawling under the impressive Molyvos Castle; Skala Eressou with its fine beach and bars; and the hilltop Agiassos, immersed in verdant forests. For your second stop, take a ferry to Chios, where you can enjoy the nightlife in the main town, and don't miss the old quarter. Travel via Lithi—derived from Alithis limin, meaning "true haven," which is rather apt for this beautiful fishing village—to enjoy a good fish lunch. Next, go to Pirgi (famous for its unique mosaics) and Mesta, part of the "masticohoria" or mastic villages, world renowned for their cultivation of mastic trees, which preserve a Greece of centuries past.
If you have less time and want to visit just one island, take the ferry from Piraeus directly to Samos. Circle the island, stopping at its lovely beaches and at Pythagorio, the ancient capital, or the temple at Heraion, one of the Wonders of the Ancient World. Consider visiting the popular traditional fishing village of Kokkari, which has managed to keep its architectural authenticity, then head for the beaches Tsamadou and Lemonaki, where the green pine slopes meet the cobalt blue waters of the Mediterranean. If you're drawn to the shores of Turkey, Samos makes a convenient stopover, as there's a daily ferry service to Kusadasi.