SA(6) and I began to read Proverbs together at the beginning of Grade 1. He had been reading for about half a year at that point, and had just received his own Bible. I decided we would try to read a short passage every day after lunch, where he would read one verse, and I would read the next.
I should perhaps mention that we have a habit of reading Scripture aloud after meals anyway, whether I or my husband do so, or whether SA reads along with us. So this new habit was actually not a huge innovation on our part.
Still, it was a challenge, and we did not really begin to do it steadily every day until the beginning of the second term of Grade 1. For one thing, the vocabulary of Proverbs is really challenging. For another thing (and this is probably more to the point), any new habit requires committed, daily effort. I do have the blessing of a son who, once he makes a habit his own, never deviates from it. You could even call him a bit inflexible sometimes. However, in this case it has been a blessing. :)
As we've really gotten into the routine of this, I'm enjoying it more and more. I'm sure a case could be made for beginning with an easier Bible translation with young readers. We've chosen to simply use the version we normally use as a family and plan to keep using throughout our children's growing up years. (ESV) We do this to build our children's familiarity with the language of Scripture over time. Switching versions as they grow could take away from that.
Beginning with Proverbs was a personal decision, and not really one I put a lot of thought into. However, it has proven to be a good choice. The best thing about it is the repetition of words. While the vocabulary may be difficult, once a child has mastered a few big words, the same words keep coming up. (Righteousness, instruction, knowledge, reproof, prudence, integrity, understanding, discretion...) There is also a lot of repetition of ideas. Comprehension may or may not be complete at this point, but it would be hard for him not to get the idea that wisdom is good and foolishness is bad, that hard work is good and laziness is bad, that listening to instruction is wise and despising it is foolish.
We are now working through Proverbs 18. It takes us about three reading sessions to get through a chapter now. It took many more than that in the beginning, but there was no rush. I do not make him "sound out" big words when we're doing this. If he's struggling, I simply supply the word and we move on. I still help him quite often, but I notice that he usually reads a word effortlessly the second or third time he encounters it.
I don't know where we'll go once we're done Proverbs, though I am leaning towards the Psalms. I anticipate that as our children grow, we'll simply incorporate each new reader into our routine until we're all taking turns reading verses aloud. I have a distinct memory from my childhood of reading like this with my family...my father, my mother, myself, and two younger siblings. The youngest reader at the time was about five years old. It is a good memory. I'd like to pass it on to my children, and perhaps to you and yours as well.