The ferry to Samos leaves from Kuşadası's main port and crosses the Mycale Strait in about ninety minutes. It is one of the easier ways to hop between Turkey and Greece: two operators run daily services in summer, less frequently in shoulder season, and not at all on some days in winter.
The Greek port of Vathy is a pleasant working town with seafood tavernas right by the quay. With an early-morning departure it is possible to do Samos as a long day trip from Kuşadası, but you'll need your passport, you must check in an hour before departure, and you'll be paying small port taxes on each side. The mountain villages and beaches of Samos are best with an overnight stay.