Captain reported a failure to follow SOPs and communications procedures caused the aircraft to roll forward under engine power when the tug and tow bar were disconnected.

Narrative

Push crew disconnected tow bar from aircraft without telling me to set brakes or hearing my response causing aircraft to roll forward approximately 15 feet under engine power. Ground control cleared our flight to push off of gate. I advised the tug driver that brakes were released and they were cleared to push. During the push back we were given clearance from the tug driver to start engines. I responded that we were starting engines. As the second engine started; the tug and aircraft came to a stop. After a few moments i felt forward movement and saw that the aircraft and the tug moving forward. It appeared we were now being pulled forward by the tug. I quickly realized the tug was pulling away and the aircraft was rolling forward. I stopped the aircraft immediately. The tug continued back to the gate. I called operations on the radio and advised what had happened. I also had the tug driver reconnect his headset to ask why this happened. I was told that he was having trouble with his headset and I must not have heard him tell me to set brakes. I had a brief conversation with him explaining how we narrowly avoided an incident and did advise him that he has to hear my acknowledgment before disconnecting.This resulted from the tug driver not following SOPs. The tug driver has to hear me say brakes set; cleared to disconnect interphone before disconnecting. There's lots of new ground crew working around aircraft for the first time. I recommend more training that stresses the importance of following procedures.