Is there a button that pilots push to transmit on the radio?
There is a push to talk (PTT) button located at the yoke or sidestick, in small and large aircraft likewise. When the PTT button is not triggered, all communication is routed through the intercom and not broadcasted on the frequency. Once the button is triggered, communication is broadcasted on the active frequency.

In the below picture of a glider, it is located on top of the middle stick.
On the second picture is it behind the yoke, to be used with the index finger.

(Image Source: WikiPedia)


(Image Source: Private Blog)