There are two options:
Option A - You have to go into Athens first. There are two main bus stations, the X93 bus going to Kifissos station is the one you need, But the Delphi bus goes from Liossion station, ask the driver to tell you when to get off.
This bus will take you about an hour, the ticket is 6 euros, kiosk outside the airport terminal, validate the ticket inside the bus (date & time stamp)
At the bus station you will get a numbered seat on your ticket to Delphi, they are good air conditioned coaches with luggage storage underneath.
Here is the website of the KTEL buses of the Fokis prefecture
The whole trip is about 2,5 hours.
Buses are leaving from the terminal that is located at Liosion Street, here is the address and tel of the Athens office at the Terminal
Liosion Street 260, 104 45 Αθήνα
Tel : 2108317096 or Mobile phone : 6936648239
Best way to go to the terminal is with a Taxi, your hotel can call you one. Calculate 20 minutes from the centre to Liosion Street.
The buses are very reliable and very modern.