This is a bit trickier than you would think, because the message alert tone settings are not in the same place as the phone's ringtone settings. In order to change your message alert tone, you can take the following steps:
From the main screen, select Settings.
You will see the options Normal, Vibrate and Ring, Vibrate Only, Silent, and Flight Mode.
Choose the profile you want to change.
Select Edit on the lower lefthand corner of your screen.