AppleScript: Index of substring in string

The built-in offset command should do it:

set s to "Today is my birthday"
log text 1 thru ((offset of "my" in s) - 1) of s
--> "Today is "

set s to "Today is my birthday"
set AppleScript's text item delimiters to "my"
text item 1 of s
--> "Today is "