For instance, did the ancient Greeks take their founding legends and the Trojan Cycle as factual? And what were the factors that allowed people to question whether a source was biased?
Always a fascinating question to explore, and continues to change each day as it takes on new meanings & consequences for us in the present day; as you are waiting for new responses/new perspectives on the question, I will attach a few related questions/answers given by users previously on /r/AskHistorians !
As a general starter, the AskHistorians wiki has some useful resources which will help exploring a question like this under its historiography - historiography being the study of historical writing itself, and the study of how the history of particular fields has been studied, and oftentimes histories of historical writing.
Contemporary Understanding of Myths
in relation to the Trojan War specifically, a deleted user wrote about the different contemporary evidence/sources alluding to it: Did the Trojan War actually happen?
/u/UndercoverClassicist tackled Did ancient Greeks literally believe in their myths?
/u/azdac7 , /u/itsallfolklore , and /u/bluecatsghost weighed in on: Did ancient/medieval people truly believe in mythological/magic beings?
/u/toldinstone made extensive contributions to: In Greek mythology, the gods are said to live atop Mount Olympus. Olympus is an actual real mountain in Greece. Did the Ancient Greeks never climb the mountain? Wouldn't it have been obvious that no actual gods lived there?
/u/mythoplokos , /u/earthvexing_dewberry , and others provided a wealth of responses to: Did the ancient Greeks/Romans believe their myths were literally true?
Some wider approaches to the question of the historical development of historical writing/discpline/methodology:
/u/AnnalsPornographie answered, in relation to western historiography, [What is the history of Historians? When and how did it become a job?] (https://www.reddit.com/r/AskHistorians/comments/9z76tb/what_is_the_history_of_historians_when_and_how/)
/u/patobie answered What is the history of historians? At what point did humans begin studying and documenting history?
/u/BoxOfMapGrids wonderfully compiled some more questions/and answers here in relation to What is the history of History?
I copy one here for ease of access:
Recent Publications
Extensive academic discourse exists on the topic - not only in regard to beginning of historical writing (and the anthropological problems to consider, such as how we define and identify historical practice in any given culture), but also in relation to problem of myths, and how we perceive a historical culture's attitude to them, and understanding of them as "real" - hopefully someone with expertise on myths and Ancient Greek culture will be able to contribute here!
J.W. Burrow's History of Histories offers an accessible introduction into the topic of the evolution of the writing of history. However, from my own reading of it, it is quite eurocentric and tends to overemphasise the western tradition of historiography (beginning with Thucydides and Herodotus) - that does not discredit the work the book does, but do bear that in mind!
A text I would recommend for anyone wanting to explore contemporary medieval approaches to historiography and from a non-European perspective, I would highly recommend Ibn Khaldun's (732-809 AH/ 1332-1406 CE) Muqaddimah. It is an introductory text to a much greater text writing by a Tunisian scholar, which explores the concept of history, the failures of historians, and the discerning of myth, fables, and lies from the seeking "of truth", and tries to understand history as a sub-discipline of philosophical inquiry. He examines the state of Islamic historiography, and provides source analysis and philological solutions to various problems he identifies (from tall tales of great cities of copper, anachronistic interpretations of individuals, and misunderstandings of the Old Testament) - predating much of the work of scholars like Leonardo Bruni and Lorenzo Valla, who are often attributed to being great innovators in this regard in Italian Renaissance historiography!