Use of Go Phone on Wifi

I am getting ready to switch my GoPhone to a newer more user friendly model. The newer model will work with the wifi in my home. I do not get a single bar of cell service in my home.

Will I accrue data charges as well as the daily charge when my phone connects to the Internet even though I am not using cellular service? Or will it cost me nothing since I using the device to go through my network not the AT&T service?

Thank you