Christian Apologetics Society has a short history of format of the original Bible, and why memorizing verses is probably a relatively recent phenomenon.
The "books" of the New Testament were actually scrolls of very expensive parchment or papyrus. The scripture was written in scriptio continua style. That is, to save space and economize, the scripture was written without chapter and paragraph breaks. In fact, there were not even spaces between words and verses as in this paragraph itself. Can you imagine trying to read John 6:53 aloud, while looking at:
While New Testament scrolls existed in the 2nd and 3rd centuries, there were few, if any, personal copies. Christians of the day did not arrive at "church" carrying their own copy of scripture. The modern Christian tradition of believers bringing their own personal Bible to church was not an early church practice. Instead the congregation assembled a library of scripture, some in the modern canon and many not found in the current canon. At their gathering, most, if not all, the scriptures were read aloud to the assembly. If an early Christian congregation had acquired a copy of Paul's epistle to the Romans, it likely would have been read in its entirety every Sunday, week after week. That's how New Testament scripture was initially memorized. The Christian tradition of reading scripture aloud in in repeating cycles is still observed in all 22 Catholic Churches and in all Eastern Orthodox Churches.
Kevin Vost has written a book titled Memorize the Faith! (And Most Anything Else) about a method of memorization devised by St. Thomas Aquinas.
Once you have memorization mastered, you can get started on Patrick Madrid's 150 Bible Verses Every Catholic Should Know.
Or get to work immediately with this free list of verses.