One of the aspects which united the Greeks of the Archaic and Classical period was a shared religious understanding. By "religion," here I mean a shared mythological substrate and a shared repertoire of ritual. Within that framework, local variation was plentiful. Every city had her own particular cults to particular deities, but generally the mechanics of those cults were similar, and the temples or structures associated with those cults were similar. At Sparta, the most important figure was Artemis Orthia, while at Athens, Demeter and Kore (Persephone) were unusually emphasized. But both cities had their own versions of Athena in her guise as the Protectress of the City. At Argos there was a major sanctuary to Hera, while at Athens, Hera did not play a prominent part. All of the major Greek deities were typically represented in all the cities/sites in some form or another—it was the magnitude of their representation that varied. Artemis at Sparta and Ephesos, Hera at Argos, Zeus at Olympia and Nemea, Apollo at Delphi. At Athens, we know of important cults to Poseidon (at Sounion, in the south of the territory), to Demeter and Kore (both at Athens and at Eleusis in the north of the territory), to Zeus, to Athena as Victor (Nike), as Protectress (Polias), and as Battle Maiden (Promachos), to Hephaestus (at the Agora), to Pan, to Dionysus, to Hestia, to Artemis at Brauron, and on and on. There is no reason to think that Athens was special in her variety—we just have a lot more evidence for Athens than any other Greek city.

Every Greek city had their own idiosyncratic cults, some of which were so old and so obscure that they defy coherent study. At Athens, the Thesmophoria festival to Demeter and Kore almost certainly contained ritual elements which were older than the Indo-European peoples who pushed beyond the Balkans and became "Hellenes." There was no Thesmophoria at any other Greek city. Observance of local cult often generated local myth to accompany the cult and to provide eteological explanation. The Greeks were very good at adapting local belief into the larger fabric of Greek "religion." The gods' epithets were good for this. Athena Promachos was not the same as Athena Polias, and this was no problem. Artemis Orthia was Sparta's gal, but not the same Artemis at the great temple at Ephesus. At Ephesus, the goddess was originally represented in the Archaic period by a xoanon, a wooden effigy which was carefully and ritually clothed, but at Athens, the same ritual xoanon was not Artemis, but Athena Polias, and the privilege of clothing the effigy every year was reserved for the daughters of the Athenian elite. And by the way, the Greek colonies in the west were not much different than the old cities of the homeland. Their religion was the same, sometimes with some local Sicel or Italian or Gallic flavor.

The religious personnel of Greek cities were generally similar. Again, we know the most about Athens, and there is no reason to think it was much different in other cities. Priests and priestesses were sometimes hereditary positions, often passed down through families. The Athenian clan of the Eteobutidae always provided the priestess of Athena Polias. From the dimmest days of the Greek "dark age," their daughters had served in this role, first in the long-lost temples of the old Acropolis, then in the shining Erechtheion. From what we understand, the generations of priestesses served in this role for eight centuries or more, until the Roman general Cornelius Sulla sacked Athens in 86 BCE and burned the Acropolis. Other religious positions were elected at Athens, or chosen among the oligarchs in other, less democratic cities. Athens had the post of archon basileus "King Archon," who had a religious function based, again, on tradition so old it cannot be traced. The Greek archons were, as a generic category, a lot like Roman consuls, who also had a combination of civic, military, and religious aspects. Sometimes it isn't clear how priests or priestesses were chosen. We have a vast amount of documentation about temples, because temples tended to accumulate lands and then rented those lands out. Rentals generated documents, and a lot of those documents survive. In them, we see a kaleidoscope of miscellaneous priests and such, most of which are very hard to track over time and space.

Secondary sources for Greek religion (from these you can get at primary sources; especially the Kearns sourcebook):

Bremmer, Greek Religion (Oxford, 1994)

Burkert, Greek Religion (Oxford, 1985)(German, transl. avail)

Easterling and Muir, eds, Greek Religion and Society (Cambridge, 1985)

Kearns, ed. Ancient Greek Religion: a sourcebook (Blackwell, 2010)

Larson, Understanding Greek Religion: a Cognitive Approach (Routledge, 2016) (a cognitivist approach; I have not read it through)

Mikalson, Ancient Greek Religion (Blackwell, 2010)

Parker, On Greek Religion (Cornell, 2011)

Ogden, A Companion to Greek Religion (Blackwell 2007) (this is a basic intro to the subject with separate articles on various topics; it is available online from most university libraries).