This question, as asked, is a little difficult to answer, so if you don't mind I might unpack it a little first. An Ancient Greek citizen, by most definitions, could be anyone from about 800 BC to 32 BC, or arguably even later. This is quite a long time, and things relating to what we'd call religion did alter over this period. However, I will do my best to give you an answer relating to that.

Also, you'll find that comparing our notion of 'religious' to the concept of piety and religiosity in pre-Christian Greece (and pre-Christian Rome) is nearly impossible, and that might help to answer your question.

Belief is not the important element of Roman or Greek polytheistic spirituality. There are certainly specific Greek and Roman authors, usually philosophers, whose work displays something we'd call similar to non-belief, or certainly highly unorthodox notions of what Gods actually were. However, worth noting is that these authors would still often refer to the traditional Gods to describe certain universal forces, and would still have the traditional dedication to the Gods at the start of their works. Even if you didn't 'believe' in the Gods, the norm seems to have still been to view the world through that lens, and to still refer to them with respect.

Instead religion was integrated into daily life and culture, and what became most important was participation. Every Greek polis community would have its own festivals and sacrificial calendar, its own particular religious traditions, sanctuaries, shrines, and temples, and taking part in these was considered to be part of being a member of those communities. The most famous ancient Greek cultural events, like the four great Athletic games (Olympian, Isthmian etc.), and the Athenian Dionysia which developed the concept of drama, were explicitly religious, being dedicated to various Gods. You can argue that from a modern perspective this is simply lip service for a secular event, but there is no concept of the secular in ancient Greek society, or many other societies prior to the emergence of great evangelical faiths. Belief may not have been a large element of these great festivals and gatherings, but religiosity was, as they understood it. This might not be the same phenomenon as faith but it had a similar strength within their society. Attacks upon sacred structures attached to a community, such as the Hermes in Athens, were considered to be attacks on that community itself. Likewise, participation in religious activities considered to be Greek was, at least from the 5th century BC onwards, part of what was considered to make a person Greek.

When it comes to the idea of the Gods as myths, there is a crucial difference between the Greeks/Romans and ourselves; we have access only to whatever preserved material exists that tells stories about particular gods and events of the legendary past, whereas they would have been immersed in a much more complete literary tradition, not to mention even more deep rooted oral ones. Even those preserved myths we do have often exist in contrary interpretations, or present radically different views of a given god. On the one hand Ancient Greek religion as a whole did not possess dogma, and set scripture, and this gives a lot of room for very different interpretations of Gods, but on the other hand there was a general orthodoxy, an idea of religious practices which were not Greek, and there were arguments about which authors and stories presented Gods accurately. As time went on, for example, the very stroppy and cruel portrayals of Gods as presented in the earliest Greek authors were often criticised by contemporary Greeks as false.

Your question about other religions, and about temples and shrines, are very big questions in their own right and I don't feel I can answer those properly here, but one thing I would caution is that past churches were not like modern churches, but you also wouldn't characterise those solely as superstition/tradition/culture. Temples absolutely were about tradition and culture, as are modern churches, but they were not places where things equivalent to sermons took place. They were centres of ritual, and guardians of sacred objects, and large societal institutions, but they were not places for preaching. But they were absolutely sacred spaces, and not the same thing as just being a large, old, and pretty building.