The oldest specific texts in the Bible are (probably) the songs of Deborah (Judges 5) and the Sea (Exodus 15).

Due to their poetic form, they preserve a very old form of Hebrew, and don't appear to have been altered linguistically as much as the parts that are stories told via oral tradition.

But as to the content of stories... it's very hard to say. The Pentateuch (the first five books, AKA "the Torah") has some very old content, and some fairly late content. (Moses we may comfortably rule out as the "author." These texts have many authors, all of whom write in Hebrew of a period significantly after Moses' time.) The book of Deuteronomy used to be singled out as a fairly cohesive work, but it too seems likely to have been written in layers over time (the book appears to begin several times). Most of the rest of the Pentateuch is all over the place with writers and times, and scholars disagree vehemently about the dating of the sources. The major sources are usually called J, E, and P, but they probably don't really represent single "sources" as we might understand them, but rather fragmentary traditions with different writers gradually accumulated together. Even the redaction/editing is done in different ways by different hands: Gen 1 and 2 are different accounts of Creation that differ on many points and are juxtaposed with no attempt at reconciliation. By contrast, the Flood narrative has two stories woven together, though the redactor's unwillingness to delete parts of either story is apparent in the fact that both may be seen in their entirety (leading to the odd sense in the present text that everything happens twice). In yet another method, the story of Hagar is presented in two different places as if to pretend they happened in one after another despite being almost exact replicas of each other. There are many other examples throughout the texts (Joseph and the well, parting the Red Sea, 10 Commandments, the theophany to Moses, etc.). All this to say that we know that many hands wrote these texts and at many times. They are the product of an entire culture.

It's a fascinating field but to get much more specific would entail me giving my own take on the scholarship rather than the general consensus. Hope this helps! Feel free to ask any follow-up questions.

As a related question, where did the theory that Moses wrote Genesis come from? And where did the idea he got it in a vision come from? I know Moses being the author is the traditional view of Orthodox Judaism, but how far back does that go? Is it pre-second Temple or a later assumption? Do we know when it was first mentioned, and how seriously it was critiqued by Rabbis and Christian Priests/theologians?

I'm sure some people reading my question will say "duh, it comes from the Bible", but a lot of books in the Bible give authors, or just start with "The Lord came to such-and-such and told them..." Now, Leviticus, Deutoronomy and Numbers do start with "The Lord came to Moses in [somewhere] and said to him..." Genesis and Exodus don't. No authors are listed. It's not very surprising I suppose for people to assume that Moses wrote Exodus, as it essentially recounts his life, but nowhere in Exodus does it say "And God came to Moses in a vision and told him the story of the book of Genesis", as far as I can remember. So did people just assume that since Leviticus, Numbers and Deutoronomy claim to be quotes from God given to Moses, that Genesis is too, or is it more complex than that?