There are a variety of goals you can work towards in reading scriptures with your children, and deciding yours will help make your scripture time more beneficial.
Do you want them to be familiar with scripture stories? Then using the scripture stories for children are fine, or jumping from story to story in the scriptures. Ask the kids what their favorites are and read them. Go in order, or don't worry about the order.
Do you want them to be able to read well from the scriptures? The language of the scriptures is definitely different. If you want them to be able to read them well, you need to have them reading in the scriptures.
Do you want to be consecutive, reading every day? If that is your goal, then what you read or how much you read might not be a factor, as long as you read something.
Do you want to finish in a certain amount of time? In August 2005 President Hinckley gave a challenge to members of the LDS church to read the Book of Mormon by the end of the year. If you want to set a goal with a time factor in it, figure out how much you need to read daily and run with it.
Determining your scripture reading goals is a huge help in deciding just what direction you need to take, and also helps in deciding just what you'll read. There are a variety of options really.
When the boys were very young we started with the scriptures stories books for children. We really enjoyed that time and it helped them learn the stories in the scriptures. It got to the point that I felt like they were missing out on the power of the words in the scriptures, things like "I will go and do the things which the Lord hath commanded," so we switched to the real deal. I wish someone had told me it was okay to jump from story to story back then, because I think if we did that we would have stuck with it better. But we didn't, so it stopped being fun for the kids and we got out of the habit.
Another option is to read talks from the most recent general conference. Buy each of the kids their own copy of the conference issue of the Ensign, and choose a day or two each week to read from it.
Use your scripture reading time as an opportunity to teach them about the scriptures. Teach your children to use the topical guide and read by topic, read from the Bible dictionary, or learn the songs that teach the books of scripture in order. Some neighbors of ours are reading from the True to the Faith book and discussing it with their children.
Now that some of our kids are older and some young, I've had to do some reevaluating. We read from the scriptures together as a family in the morning. The two youngest are in and out, and don't listen to much of what we read. I finally realized that I need to start from the beginning with them too, so I've been reading to them from the scripture stories books at night. They LOVE it.
To help children be familiar with the scriptures, draw a simple picture next to the stories so they can find them on their own, like a flag next to the verses about the title of liberty, a sheep next to the story of Ammon, and a bow and arrow next to where Nephi breaks his bow.
Next: how it "looks" in our home.