We just spent a week in Italy.
My wife has an iPhone and read many horror stories before we went. She called AT&T and had a rep tell her what to do (and she kept a record of names just in case)
Basically, she switched off data and switched mail search to manual. She really only wanted it for the camera . . .
Tell people not to call or text you, as you pay even when messages go into the answering service.
I don't know all the details because I only have a dumb phone and left it at home. I'll try to remember to let you know if we get any out-of-line charges.