There is a small bus going to Azov regularly from the bus stop near the train station in Rostov-on-Don. A sign in the front window of the bus will show "Ростов - Азов". The trip costs only 67 Rubles (April 2016) and will take an hour drive through the south of Rostov's environment. The "central" bus stop in Azov is the final stop and can't be missed.
From the bus stop in Azov further buses to smaller places near the Don are available. Kagalnik (Кагальник) is worth a visit for a beautiful trekking tour near the Don Delta.