The process of memorizing the Quran is a journey that should be relished as it builds the Muslim personality. Memorizing properly for this end in mind, should be done in a proper way so that we benefit from the process.
There are some prerequisites to begin. The foremost is to do it out of awe (tawqa) of Allah and in doing so to give up sins. It is not possible to memorize while continuing to sin. To memorize the Quran, we must love it and have a special relationship and companionship with it, by reciting a minimum set amount daily and engaging with it as much as possible.
To benefit from it, it is highly recommended to recite it as Angel Gabrial (AS) taught it to Prophet Muhammad (SWAS) and that methodology has been passed down through certifications to our present day. Also, we need to understand what we memorize, by understanding the language it is in, i.e. Quranic Arabic which is a subset of Classical Arabic, which even many modern Arabs also do not understand well today.
We cannot memorize the Quran if we do not schedule it as a daily appoinment at the same portion of the day. Thus, we need to set a dedicated time for it and seriously honour that appointment by fixing it in our schedule. As with time, the place needs to be fixed. It should be a place free from distractions where we memorize daily. As with time and place, the mushaf (copy of the Quran) needs to be fixed as well. It should be a standard copy and the same copy should ideally be used throughout the memorization.
Before memorizing a new section of the Quran, we need to write it down by hand, with the paper to be shredded after memorization is accomplished. Finally, memorizing for Allah's pleasure requires making continuous supplication to Him to facilitate the process for it cannot happen unless it is sincerely for Him.
Once the right conditions are established, we need to become familiar with the whole Quran, by reciting it frequently before beginning the memorization process. It should begin with the shortest chapters at the end of the Quran (part 30) for quick wins, and boost to confidence, as well as the fact that these chapeters are easy to recite in daily prayers by memory. Thus, we can proceed from part 30, to part 29, to part 28. Then, we can focus on memorizing the chapters that we are most familiar with like Al Khaf that we have been reciting every Friday, or Al Rahman, Ya Seen, etc. This will be easy to do and will set us up to memorize the whole Quran from the begining going forward (i.e. from Al Baqara onwards).
The way to memorize is to take a section a day to memorize. Say we start with memorizing 3 verses. We do this by reciting the first verse 40 times without mistakes. Then we recite the second verse 40 times without mistakes. Once we have recited both individually 40 times, we recite the first with the second in combination 40 times without mistakes. Next, we memorize the third (last) verse 40 times without mistakes. Finally we recite the three verses of the section in combination 40 times without mistakes.
Once we have memorized the daily section, we need to review what we have memorized before with this new section (3 verses). The reviewed section can be a quarter part of the Quran. After this is consolidated, we need to examine the section to memorize for the next day, prepare it and write it down to work on the following day. In the preparation, we need to read the translation and tafseer of that section we want to memorize.
Memorization should be done 6 days a week, with one day dedicated for only revision in which all memorization should be reviewed. By using our free time for the wird of the Quran and reciting what we memorize in prayers in the day and specially at night, the memorization gets consolidated and part of our being and that is how the Quran transforms us.