>> I have heard that you can incur charges with ATT when people leave you voicemail on the phone as you pay for it. <<
That won't be true as long as you don't call up to listen to the voice message.
Below I'll just expand on what RosemaryNJ has already said,
1) Turn Data Roaming OFF - so that your cellphone cannot connect to a 3rd party cellular carrier.
In iPhone5 this is via Settings --> General --> Cellular ---> Data Roaming -->OFF
2) Turn Cellular Data OFF - this will turn off cellular data, forcing it to use WIFI or else don't connect at all. (Voice and text message will still be on.)
For text message and voice, if you don't reply or answer, you won't be charged the international fee.
In iPhone5 this is via Settings --> General --> Cellular ---> Cellular Data -->OFF.
3) Put your phone to Airplane Mode - this will turn off both voice/text and data. So you may not need to do this if you have already done both (1) and (2).
In iPhone5 this is via Settings --> Airplane Mode --> ON.
An inexpensive alternative is to sign up for an International Text Message plan. With AT&T, and assume you have already had a regular text message plan, then it is:
50 TXT messages for $10 (only what you sent out count toward this.)
200 TXT messages for $30
If you do not already have a text message plan, then the price per text message is higher, but still very economical and convenient. You can receive/sent text message regardless whethere you have access to WIFI. This was very convenient when our group of 8 was in Bermuda last week, and needed to see if other members were in the same vicinity to meet for lunch etc.