There is no question that divination was a significant part of life in the ancient Greek world. Decisions were made at the highest level based, at least in part, on the work of oracles and other diviners.
The oracle at Delphi is probably the best known of the major oracular sites; it belonged to Apollo (following Themis, who herself followed Gaia) the Pythian (named so for his mythic defeat of Python to gain the oracle for himself), and the priestess who served there was known as the Pythia.
Other well-known oracular sites included the Apollonian oracles at Didyma, Delos, and Corinth; non-Apollonian oracles included Dodona, where the oracle belonged to Zeus and Dione, and Olympia, also an oracle of Zeus.
This is one of those areas where ancient practice comes up against the modern world. The ancients trusted the major oracles (most of the time) because they were long-standing institutions; they were therefore believed to be proven and reliable. (There were other oracular services available, individuals who operated in a less official manner and who were considered less proven and reliable because of their lack of institutional association. Credibility was thus based on factors other than ability.)
While there are a number of individual inspired oracles active in the modern Hellenic community who provide oracular services, there is no general or central oracular authority. If you seek the advice of a modern oracle, you’ll need to do some research first to find one.
Other ancient diviners relied on observational skills–the patterns seen in natural phenomena such as the actions of birds, or in the entrails of a victim sacrificed for divinatory purposes, could be read by a knowledgable seer; the reliability of this person was often not determined until after their prediction had (or had not) been realized.
Perhaps more useful for many of us is the fact that oracles that did not depend on a seer (but could be used by anyone) were popular as well, and that there are some surviving records of these. An “oracle by lot” or cleromantic oracle is one where the seeker relies on chance for their answer, rather than the interpretation of a priestess or other professional diviner. One sort is that of the alphabet oracles; runes and ogham as used by modern pagans are alphabet oracles, and there are a number of examples of ancient Greek alphabet oracles used by modern pagans as well.
An oracular city such as Delphi or Olympia would draw far more seekers than could reasonably make use of the main oracle itself. For this reason, such cities tended also to draw a variety of other diviners–seers and soothsayers with less official credibility but greater accessibility. Several of the “oracles by lot” and alphabet oracles were found in these cities as well–an additional divinatory option for those who might be unable to approach the greater oracles, and an option for the modern seeker as well.
Here are a couple of good resources for folks who’d like to give the old-school divination a try:
The Olympian Oracle by Apollonius Sophistes: This alphabet oracle is one found in the Greek city of Olympia. The site provides excellent instructions for utilizing it in three different ways–using drawn alphabet tiles, thrown knucklebones (astragali), or thrown dice–and provides both the text of the oracle and some suggested interpretations.
Astragaloi: Greco-Roman Dice Oracles by Jenna Mortensen: Very good piece providing good historical information on dice oracles in general and how to use them; also provides detailed information on one specific dice oracle from the city of Termessos; in this oracle, different throws also include reference to particular deities.