Assos (Behramkale) is about 80 km north of Ayvalık and one of the most beautiful ancient sites on the Turkish Aegean. The Temple of Athena on its clifftop hill dates to the 6th century BC; the village below has a small harbour with stone-built restaurants. The drive through olive groves and pine-covered hills is worth doing at least one way.
Bergama (Pergamon) lies 55 km to the southeast and is one of Turkey's most important ancient sites — the acropolis of the Pergamon Kingdom, with a vertiginous Greek theatre cut into the hillside, and a separate Asclepion (ancient medical centre) in the valley below. The Bergama Archaeological Museum in the modern town holds finds from both sites. Half a day is enough if you're selective; a full day allows for both the acropolis and the Asclepion. Regular dolmuş services run between Ayvalık and Bergama.
Lesbos (Mytilini), Greece, is reachable by seasonal ferry in about ninety minutes. The island has its own olive oil and ouzo traditions, several attractive stone villages, a decent archaeological museum in the capital Mytilini, and beaches on its southern coast. A day visit is feasible but an overnight gives more time — check visa requirements and ferry schedules, which vary by season and year.