Remove SIM card form ATT iPhone 4 move to 5?

Would I be able to, on my own, move the SIM card from an ATT iPhone 4 to a new ATT iPhone 5?

Yes but you will need a sim card cutter so it is the size of micro sim. Here is where you can buy it: http://www.amazon.com/...B009ERXEDK

No. The iPhone 5 has a smaller sim card than the iPhone 4. The iPhone 5 has a micro sim card. You need to contact ATT and get a sim card for the iPhone 5.