You'll need to tell your phone to connect to the Disney WiFi. There is free WiFi at both the resort and in the parks, but there are different names for the connection points. You may have to set up WiFi connections at various locations around WDW until they're all programmed into your phone.
I turned off my data connection to ensure my phone was always using the WiFi. Otherwise, I noticed that we would occassionally enter a location where I hadn't configured the WiFi connection, and my phone had reverted back to sending data through our Verizon account. (We were uploading a lot of photos, and I didn't want to use up all of our monthly allocation.)