Airplane mode disables your phone - no calls, no texts - just like when you're on an airplane. You can turn off data roaming by going into settings. In my phone (Droid 3), it's then under "Battery and Data Manager" then "Data Delivery". If you want to use your phone, you can't have it on airplane mode.
On another note - eee gads, it's amazing how little these Verizon reps know.