I have read to my godchildren from the Bible. I pick and choose what I read. I think it's important that from a young age they get to know God, and realize that they are not self-sufficient. Telling them about the creation of the world, Noah, Moses, etc. gives them a chance to relate their present with the Biblical past. I certainly don't go in to any grizzly details, nor do I explain "why" God was so angry with Soddom, other than telling them that He was angry because they didn't listen to Him.
All the details will get filled in when they are older and can understand it better. However, the core, the foundation, should be laid when they are young, otherwise, they will grow up not knowing God, and thinking they don't need Him.
I have a book that I purchased in Ukraine. It has the most lovely line drawings in it. The book focuses on the images. The text (in Russian) along the bottom simply identifies the scene. I have used this on numerous occasions to explain the "event" in my own words, with the graphic to assist. A picture is worth a thousand words.
I also make my nephews read the Sunday Gospel for the day, on the drive to church. They know how to look up which Gospel will be read, find it in the Bible and read. The Gospel in church is read in Ukrainian, and they don't understand the language well enough to understand the meaning of the Gospel. By having them read it in the car, we have a chance to "dissect" it before they get to church, so, they are not as "lost".