You can also go to settings -> cellular data -> then turn that off.
It turn off all cellular data and will restrict it to wi-fi, including email, web browsing and any push notifications.
By having cellular data off, essentially you have an itouch.
No, that's not quite right. If you turn off cellular data but leave the phone out of airplane mode, then it's true that data functions will only use WiFi, but it still will use cellular functions that do not use cellular data: specifically, voice calling and SMS. Since Verizon charges $0.05/text for inbound SMS when roaming internationally, those messages will still arrive and still accumulate charges even if cellular data is turned off. Also, any inbound calls that are received, not answered, and go to voice mail may or may not incur a 1-minute roaming charge (this can be prevented by turning on unconditional call forwarding to voice mail, but that still won't stop inbound SMS from arriving if not in airplane mode).
The only sure way to avoid roaming cellular charges on Verizon is to put the phone in airplane mode, then turn WiFi back on if you have a WiFi hotspot you want to use (including the first 50MB of on-board WiFi, if they're still offering that).