Reading Qur’aan during Ramadaan is one of the best and most virtuous of good deeds, because Ramadaan is the month of the Qur’aan. Allaah says (interpretation of the meaning):
“The month of Ramadan in which was revealed the Qur’aan, a guidance for mankind and clear proofs for the guidance and the criterion (between right and wrong)”
Jibreel used to come to the Prophet (peace and blessings of Allaah be upon him) every night in Ramadaan, and study the Qur’aan with him. Narrated by al-Bukhaari, 5; Muslim, 4268.
Al-Bukhaari (4614) narrated from Abu Hurayrah (may Allaah be pleased with him) that Jibreel used to review the Qur’aan with the Prophet (peace and blessings of Allaah be upon him) once each year, and in the year in which he died, he reviewed it with him twice. From this it may be understood that it is mustahabb to read Qur’aan a great deal and to study it during Ramadaan.
See also question no. 50781
It may also be understood that it is mustahabb to complete the Qur’aan (during Ramadaan), because Jibreel (peace be upon him) used to review the entire Qur’aan with the Prophet (peace and blessings of Allaah be upon him).
See Fataawa al-Shaykh Ibn Baaz, 11/331
Both memorizing and reviewing are reading, because no one can memorize or review except after repeatedly reading the verse several times, and for every letter there is a reward of ten hasanahs.
Based on this, it is better to focus on memorizing and reviewing.
So the Sunnah indicates that:
1- We should review what we have learned
2- We should study together
3- We should read. This is achieved when doing the first two.
In this case a person should complete the Qur’aan, even if it is only once during the month, then he should do whatever is most appropriate in his case, either reading more and completing the Qur’aan, or focusing on reviewing, or memorizing new passages. He should do what is best for him. It may be better for him to memorize or read or review. The Qur’aan is meant to be read and pondered, and it is meant to influence and be acted upon.
The believer should examine his heart, see what is best for him, and do that.
And Allaah knows best.