(SEE MY NOTE BELOW ABOUT IDE SUPPORT): Initially sending an email manually via TELNET is demonstrated and then how the ESP8266 can be utilised to replicate the process. A code example is provided on GitHub here: https://github.com/G6EJD/ESP8266-Send...
NOTE: MOST EMAIL SERVERS EMPLOY HIGH LEVELS OF SECURITY, IT IS LIKELY YOURS DOES TOO, EVERY ASPECT OF THE SETUP HAS BE PERFECT FOR EMAILS TO BE SENT!!
** CURRENTLY THIS WILL ONLY WORK with IDE ESP8266 Core v2.3.0 **
Do this: IDE, Tools, Board, Board Manager, then type ESP8266 in the top-right search box and choose the 2.3.0 release