They were very engaged with the present and with whatever was around them. This is reflected in their art (tangible figures, focus on the human form) and their mathematics (nothing abstracted to the point where it ceased to touch reality). They also had an incredibly rich and strange mythology whose various stories show up all throughout the western canon.
Different Greek philosophers believed in different things. The earliest recorded ones, Thales, Anaximander, Heraclitus, and Parmenides were sort of proto-materialists in that they were the first to break with the mythopoeic tradition of interpreting the world in favor of explaining it through reductionist, non-anthropomorphic theories (all is water, all is fire, all is one).
Then Socrates came along (as recorded/embellished by(?) his foremost student, Plato) and began to undermine society and corrupt the youth with critical inquiry. In Socrates time, there were many sophists (the term has become pejorative), wandering """"philosophers""""" who taught youth how to win arguments, and Socrates, who worked for free, lived contrary to this tendency. It intrigues me that this bullshit came before real truth seeking... anyway:
Socrates believed in many affirmative things, but from what little I know of him I think that the key characteristic of his life and work (at least, what got him killed) was the imposition of reason (harsh, inflexible, irrefutable reason) onto the naive beliefs of the people of Athens, and the resulting chasm of negativity that opened up in the center of Greek culture.
Today, in the west, everyone has gotten immunized in a sense, and we can tell when someone is leading us into a contradiction. But in those times, nobody knew what the fuck was going on, and Socrates' simple, yet logjam-producing questions (his strategy was to ask you/lead you on with simple questions until you contradicted yourself) produced such butthurt in the confident chads of Athens that he was sentenced to execution.
Notably, Socrates' friends begged to spirit him away to some other city (which they were capable of). But he took the death sentence on his own head. The reasons he gives are stirring.
I would really recommend you read some Socratic dialogues. There are guides online that show you which ones are best for beginners. They are the foundation of the western philosophical tradition. As A.N. Whitehead said (you'll be seeing this quote a lot if you read further, btw): The entire western philosophical tradition is a footnote to Plato.
The length of this response is proportional to my own igorance on the subject. I talked blindly and randomly in the hopes that someone who actually knows what they're talking about would step in, correct me, and rescue you.
