OK try the steps mentioned below. Quick and easy try different sim (than the AT&T you used prior) before proceeding with the steps below.
1. Place the non-VZW sim (sim needs to have service) back in the phone and go to settings... general... about... and wait to be prompted to update your carrier settings. If not prompted see 2.
2.Plug the phone into a computer running iTunes with the non-VZW sim and iTunes should prompt you to update the carrier settings. If all else fails try 3.
3. Restore the phone to factory settings (backing it up 1st via iTunes). Place the non-VZW sim in the phone and verify if it activates or is able to make calls. If it is simply restore the backup to the iPhone while leaving in the non-VWZ sim.