Another great idea to try to get kids to read the Bible more and to have a greater understanding of God's plan for His people than just cherry picking verses to fit their current emotional state is to provide some Bible reading punch cards. They can be used for kids of any age and when so many are completed (or the list of books has been read) they can be rewarded by a certificate handed out during church and/or a gift card to a Christian book store. I'm not sure I like the idea of winning prizes for reading the Bible but I do believe in taking time to recognize the accomplishment.
As with the Bible Reading Challenge, if you want them to think about certain things while they read you'll want to have some simple reading guides to go along with the reading or you can have a brief conversation with them to be sure they're actually reading and that they're understanding what they're reading. The guides should be simple and/or short but are necessary because kids often read the Bible (and many other things) without actually understanding. Kids don't always connect all the dots on their own or in the right order so the purpose of the guide is simply to make sure they "hear and connect" what they read.
I suggest you have numbered cards so they read in order, explaining how nobody just picks and chooses paragraphs out of a book and understands the whole story. I would suggest using The Story. There is a version for kids ages 4-8 called The Story for Children, a version for older children called The Story for Kids, and also The Story for Teens. Some of the books have discussion questions included as well that will benefit parents in bringing the story of God's people together for the kids. If you'd like to use my punch cards look here.