I understand that the main ceremonies in Greek temples were animal sacrifice. What would be an example of a prayer done during one of these sacrifices? I am also curious as to what other purposes a temple would serve.
Greek prayer usually followed a basic outline:
A great example is the prayer Achilles makes in the Iliad book 16:
"Pelasgian Lord Zeus, who live far off, ruler of wintry Dodona, surrounded there by your Elloi, priests and interpreters with unwashed feet, who sleep on the ground, you who have heard me before when I prayed, who have honoured me by striking hard at the Greek army, fulfil my prayer now. I will stay here by the beached ships, but I am sending my friend with a host of Myrmidons to war. Grant him glory, far-echoing Zeus, and fill his heart with courage, so that Hector may know my companion’s skill in war, that his invincibility does not depend on my presence in the field. And when he has rid the ships of the foe and their battle-cries, let him return to the ships resplendent in my armour, he and his men unscathed by the close combat."
As for the temple's use: it was part of a larger organizational idea, the sanctuary. The sanctuary included the temenos (the area set aside for the god and separated from the rest of the landscape), the temple (if there was one), and the altar (if there was one). It was at the altar that the sacrificing took place, and this was always outside, never inside the temple. The temple housed the cult image of the god (for instance, the giant image of Athena Parthenos inside the Parthenon). It also sometimes had rooms to store votive offerings. In the Parthenon, this was a separate room and was entered from a different end of the building than the cella where the cult statue was located. Some temples served multiple cults/gods and could have different subdivided areas for each. Most major sanctuaries were staffed by priests or priestesses; in many cases, these posts were hereditary. For instance, the Eteoboutadai family always supplied the female Priestess of Athena Polias at Athens, who served the goddess at the Erechtheion. There was another priest for Athena Polias, in the Parthenon next door. If the sanctuary had a priest or other personnel, it was their job to protect the place and keep people out. Most temples were locked and were generally not open to the public except on special days. It was nothing like a modern day church in Rome, for instance. If there was a priest, a regular guy could not just wander in and sacrifice on the altar. Everything was carefully regulated by the priest(s).
In addition to all this, Greek sanctuaries from the very earliest periods were often utilized in 'secular' ways. For instance, we have a ton of archaeological evidence for group dining at temples (like at Isthmia). Some of it might have been associated with ritual dining, but it is likely that a lot of it was 'secular' and the sanctuary was being used as a conveniently-large space (like modern Alcoholics Anonymous meetings in American churches).
Large sanctuaries also often served as a place of asylum for the destitute. A certain sanctuary of Artemis at Samos had a reputation for being something like an orphanage in the Classical period (Herodotos 3.48 mentions 300 boys living there!).
Like medieval churches, large Greek sanctuaries owned considerable tracts of property to pay for the upkeep of temple business. These sanctuaries were frequently the brokers of the leasing of this land, and we have tons of data about these transactions (entire books worth, if interested). Conversely, we have a lot of preserved sacred legislation which is trying to regulate the misuse of sanctuary lands (illegal grazing, encroachment, etc).
The larger sanctuaries served as lending institutions, or surety of liquidity. The Athenians frequently "borrowed" funds from Athena during their war with Sparta, with interest, and dutifully paid it back in installments as their fortunes rose and fell during the war. When the Athenians are preparing to invade Sicily in 415 BCE, they assess the wealth of the place, and this includes not only the "civilian" wealth but also the potential capital stored up in the sacred spaces in the form of gold or silver offering objects.