a.glick, Welcome to Physics Forums!
You need to call things by their correct names! In all sciences, using the correct term is the only way to communicate ideas with accuracy.
1. Sound waves are emitted by a transmitter. Example: a speaker
2. Sound waves are received by a receiver. Example: a microphone
3. Audio can be amplified by an amplifier.
4. Sound waves humans can hear is known as "audio"
Noise cancelling headphones receive sound from the local environment. The electrical circuits inside amplify and invert that audio. That is, it is phase shifted 180 degrees. Then that audio is sent to the earphones of the person wearing the headphones. When those waves meet up with the incoming waves they should totally cancel out. This is known as destructive interference, as you said. The result expected is the wearer does not hear any sound from the external enviroment.
So, what you want to build is very similar to noise cancelling headphones, just as you said. You need to design and build a receiver, an inverting amplifier, and a speaker.