The temple of Artemis at Ephesus was one of the Seven Wonders of the Ancient World. The Artemis that was worshipped there was a little different from the image we have of the goddess as an athletic huntress, familiar from Classical art. Ephesian Artemis was represented pillar-like, with much of her body covered with what has been interpreted as breasts (symbolizing fertility, which sort of fits as she's a goddess of the wilderness); they have also been regarded as gourds, eggs, or even bull's testicles, which are all still fertility symbols). These different interpetations are discussed (with references) by S. Skarmintzos in an article published in 2017 (and available on his Academia profile).

It's important to stress that the temple in ancient Greece is the place where the cult statue of the god(dess) is housed, and where offerings are kept. It was part of a sacred precinct (the temenos), usually delineated from the surrounding area using a low wall. Priests and priestesses took care of the sanctuary, including the temple and the cult statue, accepting offerings, conducting rituals, and so on. The temple wasn't accessible to most people; usually the doors were opened during festivals and since many temples (not all!) were oriented east-west, the statue would be lit by sunlight that streamed in through the open door. Cult activities, especially sacrifices, took place outside, at the altar located within the sacred precinct, usually in front of the temple. (Compare that with a church, where the altar is located inside.) The whole complex (temenos, temple, and altar) formed a whole, a sanctuary. There were also open-air sanctuaries (without temple), sacred caves, and so on.

Everyday life at the sanctuary in Ephesus wouldn't have been too different from that of other sanctuaries in Greece, except the obvious difference in scale and number of visitors. The operations of ancient Greek sanctuaries is covered quite well by Jan Bremmer in his Greek Religion (1999), which is a great introduction to the subject. On priests and priestesses, he writes (pp. 27-28):

In the sanctuaries, priests usually officiated for gods and priestesses for goddess, but, as with sacrificial victims, there was no iron rule: Athena regularly had a priest. Priests performed sacrifices and guarded the treasures of the sanctuary, but in larger sanctuaries special personnel did the more menial jobs, such as preventing birds from fouling statues. In smaller, rural sanctuaries priests were not always present and here worshippers themselves could sacrifice after having called for the priest in vain. As mediators between gods and worshippers, priests distinguished themselves through their white or purple clothing, and on vases priestesses are often pictured with metal keys [...].

The keys were used to look doors and keep the valuables in the temple safe. Sanctuaries could collect prodigious amounts of valuables over time: at the major Panhellenic sanctuaries of Olympia and Delphi, for example, cities began building structures dedicated specifically to housing their dedications to the gods, the "treasuries". The Treasury of the Siphnians at Delphi is famous for its reliefs. Still, large sanctuaries like Olympia sometimes had so much stuff that the priests periodically cleared them out, burying the items nearby. A large number of the weapons and armour unearthed at Olympia was found in such deposits. Large santuaries also possessed large estates, which were usually protected by sacred laws, and they often functioned like banks, too (Bremmer, pp. 31-32).

At this point, it's perhaps useful to repeat the old chestnut that the ancient Greeks had no word for "religion" as such and they did not conceive of the divine as separate from everyday life. The gods (and spirits) were everywhere. Let's pick up with Bremmer again (p. 31):

Greek sanctuaries functioned in a much more varied way in society than modern churches [...]. [M]any sanctuaries contained temporary and permanent buildings which were used for dining [...]. The small Greek houses offered little possibilities for larger groups and, moreover, a sanctuary was a secure place to meet, since it was divine property.

Indeed, sanctuaries were sacred ground; people who were afraid of their lives could seek refuge at an altar or temple and assume to be protected (which didn't always happen). Sanctuaries also had to be kept ritually pure, which means: free from religious pollution (miasma) caused by e.g. childbirth, menstruation, sexual intercourse, contact with a corpse, or bloodshed. (If you suffered from miasma, you had to be ritually purified). For more, see Andreas Bendli's chapter on the subject in Daniel Ogden's Companion to Greek Religion (2007) written by (pp. 178ff). The following from that chapter sets the scene (p. 179):

By modern hygienic standards, ancient Greek cities, like all other pre-modern urban centers, were very dirty places indeed. Overcrowding and its many insalubrious effects, urban pollution, and an insufficient understanding of the necessity of waste disposal made the ancient city particularly disease-ridden.

The same Companion has a whole chapter by Beate Dignas devoted to "A day in the life of a Greek sanctuary" (pp. 163-177), which should cover exactly what you're looking for. I'll touch on a few more topics.

The most common reason to visit a sanctuary was to conduct sacrifices and to make votive offerings. The latter were used to ask for favours from the gods or to thank them. If you've been to an antiquities museum, you've probably seen votive arms and legs (reliefs, small terracottas, etc.). These were donated by people who either suffered from an affliction or had recently been cured. There are also miniature pottery vessels that were left as offerings, and so on. As Bremmer puts it, votive offerings "would be a more lasting testimony than a sacrifice" (p. 33), and because they needn't have been made of expensive materials, everyone in Greek society could make them.

Sacrifices are a particularly important feature in Greek daily life. Most Greeks didn't eat meat on a regularly basis. So when a festival or other special event was organized, the sacrifice of an animal was one of the rare occassions where people could eat meat. According to the Boeotian poet Hesiod, in his poem Theogony, the god Prometheus was responsible for how sacrifices were conducted: the bones, organs, and other inedibles were wrapped in fat and burned to the gods, leaving the mortals to cook and eat the regular meat. While Zeus wasn't fooled by the fat, and he did punish both Prometheus and the mortals, the practice continued.

The concept of the "weekend" is a modern one; in ancient times, festivals were used to celebrate and to do something other than work. Sanctuaries were an important part of festivals, which were organized frequently. People prayed at the sanctuaries, sang hymns, engaged in various ritual acts (Bremmer, pp. 38ff; note Burkert in the further reading below), such as the aforementioned sacrifice, but also dancing, as well as taking part in various contests (music, theatre, athletics), and so on. Processions were an important part of the festivities, too (and sometimes had a political significance: processions from the city to rural sanctuaries was a way of affirming the connection between the urban centre and the surrounding countryside).

Hopefully this answers your question a bit. Let me know if there's more that you want to know specifically. Recommended reading in general:

• Jan N. Bremmer, Greek Religion (1999).
• Walter Burkett, Greek Religion. Archaic and Classical (1985; English translation of the German original published in 1977).
• Daniel Ogden (ed.), Companion to Greek Religion (2007).

Another interesting book is S.C. Humphrey's The Strangeness of Gods. Historical Perspectives on the Interpretation of Athenian Religion (2004), but as its subtitle indicates it focuses on Athens.