Feel free to talk about what is being read, either during the reading or afterward. Don't make it a sermon, but short comments or questions at times are helpful. If you have a story that illustrates a teaching that was just read, the children may benefit from hearing a modern illustration of a Bible passage. Bible class should not drag on and on. If children want to talk, then you know you have their interest and can continue. If not, stop talking.
The Bible is written at an adult level, and with adult concepts. Be prepared to explain such things as sex and adultery in a way that children can understand. We find the phrase "get naked with" is a child friendly explanation for sex. Some chapters in the Bible are interesting. Some are boring, especially for children. Read every chapter in the Bible, from beginning to end, regardless of how interesting or boring. If you don't find it irreverent, you can add some interest to particularly boring chapters. For example, Numbers chapter 7 is tedious. It is a list of items each tribe brought as a gift for the dedication of the altar. There are twelve tribes, and each brought the exact same thing. This chapter is the same list of items, repeated twelve times! We have taken to passing the Bible around for each new tribe's list, and letting each child (or adult) read in the accent of choice. For example, the list for tribe of Judah might be read in a Spanish accent, the list for the next tribe with a French accent, the list for the next tribe with a Russian accent, then a deep southern drawl, a British or American accent, or speaking like an announcer, or whatever. Something like that transforms the most boring chapter of the Bible into the most anticipated chapter of the Bible.
Psalm 117 is the shortest chapter in the Bible. Psalm 119 is the longest chapter in the Bible. As a suggestion, read Psalm 117 and 118 on the same day. Then split Psalm 119 over two days. If you wish to change the Bible reading for a particular holiday, go ahead. The nativity story can be read on Christmas Eve and Christmas Day. Passages from Easter can be read at Easter. Or, keep going with the regularly scheduled Bible passages on those days. It is up to the parents.
The entire Bible course may be altered by the teacher as needed. The foregoing were suggestions and recommendations. Another recommendation is to sing a hymn before reading the Bible. The next link talks about hymns and Bible reading.
(3) Bible Reading: Things to Be Aware of (You are on this page)
(4) Suggested Hymns with Bible Reading