I agree that learning in parts is best - it is good to know, not only that there are 2 big parts, Old and New Testaments, but that each of these is divided into groups too, so as to make more "sense" when reading the whole story of the Bible.
I taught my children the books of Moses (the books of the Law) Genesis - Deuteronomy with the tune of "the more we get together".
The 12 books of history to our own made up tune, but I like the previous poster's thought to group this in 3's (Joshua, Judges, Ruth; the Samuels the Kings, the Chronicles; Ezra, Nehemiah, Esther).
We learned the 5 books of poetry to a tune that someone made up and put on a tape called "each little dewdrop", which song also explaned a little of what each book focused on (Job is all about patience, Psalms is all about praise, Proverbs are words to live by, Ecclesiastes man's ways, Song of Solomon teaches all about love...). You could also use the tune "the more we get together" again ("let us sing the books of poetry"...).
The books of prophecy can be further divided into major and minor, based on the size of the books. The first 5, I taught to the tune of "up on the housetop" (the santa song, which my kids didn't know anyway) - "Isaiah said here am I send me; Jeremiah wept in his prophecy; he wrote Lamentations and Ezekiel saw; a wheel in the sky, Daniel saw a statue fall..."
The minor prophets we learned to the tune of 10 little animals.
The Wee Sing people did write a song for the entire Old Testament, which we also learned eventually, and it's pretty good too.
The New Testament books songs are many, and I went online to find a video of the one that I use, but came across a whole bunch of videos of different versions to learn the books of the Bible too. If you do a video search on "books of the bible song", you can find several versions for the New Testament, and for the whole Bible all together.
I'm working on finding a way to learn where the books of the prophets fit into the books of history - there are a few memory tricks I've found, such as that the prophets to "O ther N ations" were O badiah and Jon ah and N ahum (Obadiah to Edom, and JoNAH and NAHum to N inevah). If you hear any, let me know. :)