The only way to get from Athens to Delphi, and back, on public transportation is by bus. There is no train line arriving nowhere near Delphi, and of course (lol) no commercial flights.
The bus line (Athens-Amfissa) is operated by KTEL Fokidas. Amfissa is a town in the Fokidas district, the end station, with Delphi in between. KTEL is a Greek federation of bus companies, KTEL Fokidas is the particular company you need to buy tickets from. You will need to start your trip from the KTEL 'B' Terminal Station Liosion, in seedy, downtown northern Athens.
You CAN buy tickets, and reserve a seat, online, unlike what you might read on the web. Just go to http://www.ktel-fokidas.gr/en/ then SELECT ENGLISH (otherwise you WILL get greek text later, in that case change the =gr in =en in the URL) Scroll down, click on 'Buy tickets' and follow directions. You can also reserve specific seats, I recommend a place on the right side for the trip to Delphi to enjoy a better view. Payment is by credit card of course and you will get (english) PDF tickets to print and bring with you. If the PDF doesn't open in Adobe acrobat, don't worry, it happens sometimes with website generated PDF. Just use Chrome or some other reader. Here's a ticket:
Choose Athens-Delphi(Station) and Delphi(Station)-Athens. You might find Delphi(Museum) wich is a stop in front of the sanctuary entrance, but it is not worth the hassle, the main station is just a few minutes more interesting walk. I recommend to take the first bus from Athens and the last bus to Delphi. This means departing from Athens a 07:00/07:30 and from Delphi at 18.30/19.00. A visit to Delphi, the museum and a the town takes its time: going up later means having too little time.
You can also buy roundtrip tickets directly at the bus station, but there's always the risk of finding a full bus.
KTEL B Terminal Station is in Liosion Street. Well, nominally. If you look it up in google maps you get..
The only problem is that the terminal is not there: the entrance in Agiou Dimitrou street, a bit up from the gmaps marker, the simplest thing is to follow Liosion street until Gousiou street and then turn right:
The terminal entrance from Agious Dimitrou is this:
To get to station one could of course call a taxi. If you want to get there by real public transportation instead just stop at the Attiki metro station (red line 2, green line 1). When you emerge from the Attiki station just look where the platform roof is:
To go to the Liosion station you must follow the direction of the vehicles running on the lane near the roof. Now you will have two options: just walk, the bus station is less than 20 minutes away, or take any street bus going in the right direction, count three stops, then check your map and walk. Bus frequency is variable (timetables are on google transit, for what it's worth), it could actually take longer waiting for the bus than walking. Just be sure to be at the station at least 20-25 minutes before departure. If you want to take the seven o' clock bus, you'll have to wake up early. Liosion street is as I said a seedy, run down part of Athens: most of the city outside the main tourist areas is like that actually, but I don't think there are real security problems with that.
The station has a waiting room with a bar, a newstand and of course ticket booths on the left entering from Odos Gousiou:
If you come there early, the bar is the only possibility of eating breakfast (all other venues will be closed). Your bus will be invariably in lane SEVEN, a the opposite side:
The only reliable schedule will be departing time from Athens. You can safely add 20-30 minutes to the nominal duration of the trip shown on your ticket. In my case both during the outward and return trip there was a 10-15 minutes stop at this 'Friendly Cafe' about 45 minutes before Delphi. It's a reasonable place to drink something and/or go to the toilet.
My bus trip was comfortable, about half of it is on a highway and then on a provincial road, in good conditions. The mountain landscape near Delphi is quite spectacular. The main Delphi 'station' is this.
For visiting Delphi, you can find lots of information online. My advice: if possible try to choose a day where the weather forecast will show temperatures below 30C, visiting the sanctuary is quite a walk. I recommend this route: from the station follow the directions, then walk past the museum and the main entrance. Then walk about 1000-1500 mt along the road, reache the Castalia spring, the Gymnasium, and finally the Athena Pronaia sanctuary with the Tholos. Start your visit there (you don't need a ticket for Pronaia, Castalia and Gymnasium) then walk back to the Gymnasium, if it's open, see the Castalia spring, then buy the ticket at the entrance and visit the site. Keep the ticket, it's good for the museum too, which I visited soon after the site.
Taking the first bus from Athens and the last from Delphi leaves enough time to see the Sanctuary without haste and then visit the museum for one or two hours. After that, there will be enough time to walk in the little town of Delphi (mostly tourist shops, but nice, and with a terrific view of the gulf of Corinth) and eat something. Καλό ταξίδι!