Att International Cell Phone Use

I live in the United States and will be traveling in the UK and on occasion, throughout Europe. I currently have ATT cell phone. I have a few questions related to effectively using a cell phone abroad:

-First, I will enable by current US phone number and SIM card to be able to be used internationally. It will be expensive when making local or international calls from the UK and Europe, but I will probably turn it on on occasion just to see if there are any voice mails or if I need to make a quick call when traveling. I was once told that if I turn the phone on in another country, even if I then turn the phone off, if someone leaves me a voice mail, even if I check it from another phone, since the last registered tower was in a different country, I would be charged for every minute a person leaves me a voice mail. Is this true? How can I get around this? Chances are I will have to turn my phone on, but I do not want to be charged for voice mails people may leave since for a few months, the last active registered tower will be a UK or other European tower.

-In addition to activating and turning on my phone with the US number and SIM card, I will also buy a local SIM card to put in my unlocked phone. Is this better than buying a go phone and adding minutes? If it is a blackberry, can I buy a text and data plan on this prepaid SIM card? What if I only buy a prepaid minute plan for my unlocked phone and SIM card; will the data function not work? Can I use this prepaid SIM card and phone in other countries within Europe?

Thank you so much for the information.