When you visit Epidauros or Olympia you can transfer the site quickly to a modern counterpart. Epidauros, the spa town. Olympia, the sports place. With Delphi this is way harder as it's not really clear how/if the Oracle really worked.
One explanation is that the Oracle would give ambivalent advice that would hold true no matter what. The most famous example is the advice "If Croesus goes to war he will destroy a great empire." True either way as you can see. Another explanation is that Delphi was like an intelligence hub. The whole Greek world would pass through. And to obtain advice, you needed to share information first. So the priests at Delphi would have access to a wide range of very sensitive information. Thinking about it, they maybe were what we nowadays call mentalists, and all they did was to use cold and hot reading techniques. Unfortunately, no historic description that would shed some light on the process remains.
The site roughly consists of three parts. The area above the road, the area below the road and the museum. All are stunning and convey how central a role Delphi played in the Greek world for centuries. The museum holds many great pieces with the horse carriage rider the greatest piece.
To me Delphi is on par with Olympia, another great site, just a tiny bit below the Akropolis as it's lacking the iconic image.
Getting There
Athens has several bus stations and I paid a Greek cabbie 15 EUR as I went to the wrong one. The correct bus station is Liosion Terminal B. The bus to Delphi is operated by KTEL Fokidas and continues to Amrissa. Travel time should be around 3h including a stop at Livadia.
In Delphi (town), the stop for the return bus is across the street from where they drop you off. The ticket booth for the site is near the museum. The main site entry is further up the road. For the lower site, you have to walk along the road (no sidewalk) some more till you get to the entry. You will pass the spring.
I had wanted to connect from Patros (doing a circle around the Peleponnes). However, I wasn't able to find a bus connection from Patros to Itea, so I returned to Athens first. Also, Hosios Loukas monastery is just around the corner, but again, I wasn't able to find a connection. And having spent substantial amounts of money on Greek cabs already and having visited Daphni monastery in the morning to get my tick, I decided not to make the effort. Obviously, if you travel by car, you should be able to make this work.