How was God's revelation to the Jews recorded?

Answer 1

The Ten commandments were given by God in direct revelation. the rest of the Torah was spoken by God to Moses during the forty years in the wilderness. Before his death, Moses wrote the complete Torah.

Answer 2

It was recorded by Moses in the Torah (Exodus ch.19-20).

According to tradition, Moses wrote a scroll containing the narratives of Genesis, at God's command, before the formal Giving of the Torah (Rashi commentary, Exodus ch.24). It was this scroll which he read to the Israelites (Exodus 24:7) as an introduction to what God's covenant would entail. These narratives were not unknown to the Israelites, since they had carefully preserved the traditions of the events of the lives of Abraham, Isaac, and Jacob (see Midrash, Shemot Rabbah 5:18 and 22).
The words of this scroll were soon incorporated in the complete Torah itself by God, including the ancient traditions of the Creation, the Flood etc. (These traditions had been known by mankind worldwide, except that among the other nations [the idolaters] they had become garbled with idolatrous drivel.)
When God gave the Ten Commandments (Exodus ch.19-20), He gave them in writing, inscribed on stone tablets (Exodus 31:18), while He taught the rest of the Torah to Moses orally, on Mount Sinai (Exodus 24:12), without writing it.
At God's command and precise dictation (Deuteronomy 1:3), Moses penned the entire Torah (Deuteronomy 31:24) immediately before his death, so that it included events that had happened in the preceding months (such as Numbers ch.20).

No Hebrew copy of the Torah has ever been found to differ with the others, worldwide. The Torah we possess today contains the exact wording written by Moses.

See also:

What is the history of the Hebrew Bible?

Some facts about Torah-scrolls