Wikipedia, a generally reliable source, reckons that the first SMS text message was sent in 1992. This sounds about right. More specifically, Wikipedia says:
SMS messaging was first used in December 1992, when Neil Papworth, a 22-year-old test engineer for Sema Group, used a personal computer to send the text message “Merry Christmas” via the Vodafone network to the phone of Richard Jarvis.
SMS took off in Europe, due to the high penetration of mobile phones. The high cost of calls meant that young people quickly adopted the technology, being prepared to get fluent with the inconvenient alphabetical mappings of the phone keypad. SMS penetrated the US and Canada much later, partly because instant messaging was already widely used. My recollection is that SMS was widespread in the UK by about 1995, but only became common in the US around 2008–these are just impressions of penetration dates, I would welcome further input.
If anyone has further information, please post a comment.