You can only choose from the text alerts provided by Apple for SMS or email alerts. If you searth the forum for changing SMS alerts there are ways to add your own but it isn't a really simple process. You can't just set a ringtone as an alert. If you have alerts set and they aren't working try a reboot. Small glitches happen occasionally.
It may sound stupid, but do you have the ringer switch off? Check your settings under Sounds and make sure you have an audible (to you) sound selected and ensure the volume on your phone is up. Try rebooting the iPhone also.