The Greek gods were their major religion and were most definitely believed in, in the same way as religions are today. As a side-note, the degree to which myths were believed in (such as the involvement of the gods in the Trojan War) would have varied from person to person. Your average farmer may have believed in Hercules etc, but we can see one of the first "historians" Thucydides, stating his aim to avoid myths (such as involvement of the gods) when writing about the Peloponnisan war.

You may be interested in past posts on this topic:

• Did the ancient Greeks/Romans believe their myths were literally true?
  ^(17 Jan 2016 | 43 comments)
  ^(/u/mythoplokos discusses to what degree belief permeated daily life in Rome & Greece.)

• Did the Greeks really believe in their gods?
  ^(02 Feb 2013 | 291 comments)
  ^([deleted] links to past comments and summarizes one of the historiographic interpretations of the matter.)

• Did the Classical Greeks really believe their myths to be fact?
  ^(01 Oct 2013 | 19 comments)
  ^(/u/XenophonTheAthenian puts Classical Greek religious belief in context with the general development of Greece.)

For followup questions, use username mentions in this thread to direct questions to specific commenters.

To add some perspective, it is worth noting that while belief in the pantheon, sacrifice, cult ritual, and oracular consultation were huge parts of ancient western culture, there were also many skeptics of religion among Greek philosophers. Thinkers like Diagoras, and arguably Socrates and Heraclitus, were atheist. Others were more moderate, but still had doubts, such as Protagoras, who considered the existence of gods unknowable. These philosophers were certainly not alone in their doubts.