Back in '72 or so I took a fantastic English college class "The Bible as Literature." The book we read, besides the Bible itself, was "The Dartmouth Bible." I don't know if it can still be found, but that would be perfect for the OP. I recall it would describe tales in the Bible along with historical background and alternate explanations for various miracles depicted in the Bible.
(as an aside, the Professor was an older African American woman, raised in Jim Crow South whose father was some sort of "Holy Roller" minister. Her passion made this a fascinating course).
What a great question from the OP.
In about 1988 I also took "The Bible as Literature" as an elective for a liberal arts requirement. The class was taught by a middle-aged Jewish atheist. Fascinating class, and the professor and I spent a lot of time talking about his course choices and organization.
IIRC, we started with some of the books with better known stories from the Old Testament's shorter books (Esther, Ruth, Jonah), and also spent some time in the book of Job.
We also dipped into the wisdom literature: Proverbs, Ecclesiastes, Psalms.
In the New Testament I would recommend starting with the book of John, and then maybe the book of Acts (many phrases from the parables make their way into secular literature), then the book of Romans.
I don't recall the version we used, but I have a 1984 New International Version that is easy to read and understand. A lot of the references in "classic" literature correspond to the King James Version, but you'll still recognize it when you come across it.
Good luck on your endeavor. As another poster said, it's easy to get bogged down in the "begats," etc. If you do pick and choose this way, a good commentary or study bible with the commentary on the same page can make a world of difference. Context as to customs and to historical events and mores aid understanding. The MacArthur Study Bible is one that comes to mind.
If you spend much time driving or prefer listening to audio commentary, there is an app called "OnePlace" that has audio from a preacher named Dr. J. Vernon McGee (1904-1988). He preached sermons "Thru the Bible," book by book, chapter by chapter, verse by verse, and I've found him to be an interesting and engaging resource. Wiki says he was an "American ordained Presbyterian minister, pastor, Bible teacher, theologian, and radio minister." It looks like there's about 30 days available at one time, and it can be interesting to read the passage then listen to him go through the same passage.