Are you testing by sending messages from a non iOS device? It's the best way to take iMessage out of the equation.
Some SMS's use the 'APN settings' as far as I recall (I think it is the 'multimedia messages', e.g. ones with additional attachments), if that is incorrect you may need to consult your carrier & re-enter the appropriate settings. Does Cellular data work for you?
About cellular data network settings on your iPhone and iPad - Apple Support
Try searching the help forums or support for your carrier with the term 'APN settings'.
Obviously reboot your device if you haven't done so already incase it is just an intermittent issue after the update.