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The way to avoid big phone bills and overage charges is to disable this feature, which comes enabled when you buy the phone. Go to Settings, open the Cellular Menu, scroll down to the bottom and disable Wi-Fi assist.
Source: I just read this on clarkhoward.com, and have done this.
You are right! Also another thing to do with a smart phone is turn off your data. We have internet/wifi at home so I always make sure my data is off when out and about.