My grandparents were immigrants: Mom's parents had no formal education, but my Grandfather taught himself to read and write in both his native language and English. My Mom said he used to read aloud to my Grandmother from the Bible. He was a bit scandalized by some of the Old Testament stories, such as the one about Lot and his daughters, but continued with the wise, old-country mentality that some things just happen, and he moved on.
My Grandparents had passed, by the time I came along, but the Bible my Dad gave my Mom for a wedding present was well-used. She would read to me- Samson's was one of the stories, as was that of Joseph and his brothers- in nightly installments.
In 13 years of Catholic school, we never rote-memorized chapter-and-verse: we were taught meanings, and that can be two different things. I've read from several versions of the Catholic Bible (I, too, have a set of Navarre!), including the Ignatius, Jerusalem and of course, the Douay-Rhiems.
I'm older now, and don't always remember as well as I used to, but then there are times when the Word will come to mind, right when I need It most!
If I can cite where a verse is, it's usually because I recall the story or context it is in. That's a big help! The rest came from the practice of my Faith within the Church: I believe what the Magisterium of the Church teaches. That really is a lamp unto one's feet! When in doubt, check with Rome!
My favorite is always Jonah I think. When I taught Bible School we tended to stick with the OT for the Bible stories and the NT for the Sacraments. But I digress. Someone came up with a way to but a bunch of black plastic together and used some fans. It was big enough for the kids to go in and experience being inside the belly of the whale. It was a great teaching tool. The little ones really got it.
That's pretty good. Chapters and verses 'sometimes' get in the way of the context. For example in 2 Timothy the latter portion of chapter 3 and beginning of chapter 4 is actually the same subject discussed. Chapter and verse helps a bunch when you are studying with others AND when one is teaching or preaching. It is a great way for others to keep up.
I have found that after quite a few years of reading the entire NT and Psalms at least 2X a year and the remainder of the OT at least 1X a year, trying to find a passage is no longer difficult. As with sports, practice of exercises with repetition leads to 'muscle memory.'
With the regular 2X NT and Psalms; 1X OT I also have specific studies, devotions (older sermons) and as I teach my sons more 'muscle memory' is added.