The easiest way to travel between Greece and Turkey is on a ferry from a Greek island to the Turkish mainland (or vice-versa).
Fast Catamaran or convention ferries depart from Rhodes, Kos, Samos and Chios to Marmaris, Bodrum, Cesme, Dikily, Ayvalik and Kusadasi.
The most frequent ferries are between Rhodes and Marmaris.
You can often get a same-day round-trip ticket for little more than the standard one-way fare, meaning that you can make the voyage over in the morning, see the sights in the port, then return to your original port in the evening.
It is recommended that ticket reservations are made in advance for ferries to Turkey!