He wants to know that hadiths that the Prophet (blessings and peace of Allah be upon him) instructed us to memorise
Question

Please write for me the hadiths in the texts of which the Messenger (blessings and peace of Allah be upon him) instructed us to memorise them.
Praise be to Allah.The Sunnah is one of the major sources of sharee‘ah, for the Qur’an commands us to take whatever the Prophet (blessings and peace of Allah be upon him) brought.
Allah, may He be exalted, says (interpretation of the meaning):
“And whatever the Messenger has given you - take; and what he has forbidden you - refrain from. And fear Allah; indeed, Allah is severe in penalty” [al-Hashr 59:7].
Hence it is encouraged to memorise whatever we can of the Sunnah, but this does not refer to any particular hadith.
It was narrated that Zayd ibn Thaabit said: I heard the Messenger of Allah (blessings and peace of Allah be upon him) say: “May Allah bless a man who hears a hadith from us and memorizes it so that he can convey it to others, for perhaps he is conveying it to one who will understand it better than him, and perhaps the one who conveys knowledge does not understand it himself.” Narrated by at-Tirmidhi (2656). And he said: Concerning this matter there are reports from ‘Abdullah ibn Mas‘ood, Mu‘adh ibn Jabal, Jubayr ibn Mut‘im, Abu’d-Darda’ and Anas. The hadith of Zayd ibn Thaabit is a hasan hadith. It was narrated by Abu Dawood (3660) and classed as saheeh by al-Albani in Saheeh Sunan at-Tirmidhi.
It is well-known that the importance of what is memorised is based on the content of the text. That which has to do with explaining obligatory duties and things that are prohibited is what the Muslim should definitely know and memorise if he can. After that, come the hadiths which explain what is recommended (mustahabb) and what is disliked (makrooh).
Hence what the Muslim is advised to do is to pay attention to learning the hadiths which explain the rulings that he needs to know, such as the hadiths which speak of the rulings on purification (tahaarah), prayer and fasting, then the rulings on zakaah, if it is obligatory in his case, and the rulings on Hajj… and so on.
One of the most useful books on this matter for the beginner is ‘Umdat al-Ahkaam by al-Haafiz ‘Abd al-Ghani al-Maqdisi, then Buloogh al-Maraam by al-Haafiz Ibn Hajar.
He should also know the proven hadiths that speak of good manners and attitude. Useful books on this matter include al-Adab al-Mufrad by Imam al-Bukhari, and one of the useful, comprehensive books on this topic is Riyadh as-Saaliheen by Imam an-Nawawi (may Allah have mercy on him). If the beginner student starts by memorising al-Arba‘een an-Nawawiyyah then the reports added to it by al-Haafiz Ibn Rajab (may Allah have mercy on him), that is good and there will be a great deal of benefit in that for him, in sha Allah.
It is good to memorise the wording of such hadiths, but if that is difficult for you, then it is sufficient to understand their meanings; commentaries on these hadiths are available, praise be to Allah, and they can easily be accessed on the Internet.
But there is a category of hadiths of which the Muslim should pay attention to learning the wording without changing it; these are the hadiths of du`a and dhikr.
It was narrated that al-Bara’ ibn ‘Aazib said: The Prophet (blessings and peace of Allah be upon him) said: “When you go to your bed, do wudoo’ as for prayer, then lie down on your right side and say: 
Allahumma aslamtu wajhi ilayka wa fawwadtu amri ilayka wa alja’tu zahri ilayka raghbatan wa rahbatan ilayka, la malja’a wa laa manjaa minka illa ilayka. Allahumma aamantu bi kitaabika alladhi anzalta wa nabiyyika alladhi arsalta (O Allah I submit my face to You, and I entrust my affairs to You, and I rely totally on You in hope and in fear of You. Verily there is no refuge nor safe haven from You except with You. O Allah, I believe in Your Book which You have revealed and in Your Prophet whom You have sent).  
Then if you die during the night, you will have died following the fitrah (sound nature of man). Make these the last words that you speak.” 
He said: I repeated it back to the Prophet (peace and blessings of Allah be upon him), and when I reached the words. “Allahumma aamantu bi kitaabika Allahi anzalta (O Allah, I believe in Your Book which You have revealed),” I said: “Wa rasoolika (And Your Messenger).” He said: “No. Wa nabiyyika alladhi arsalta (and Your Prophet whom You sent).” 
Narrated by al-Bukhari (247), Muslim (2710). 
Al-Haafiz Ibn Hajar said:
The most appropriate thing that may be said concerning the reason why the Prophet (blessings and peace of Allah be upon him) corrected the one who said Rasool (Messenger) instead of Nabi (Prophet) is that the wording of the adhkaar is tawqeefi [i.e., they can only be known through divine Revelation and sound texts of hadeeth, with no room for ijtihad], and they have special characteristics and benefits that are not subject to reasoning, so they must be memorised verbatim as they were narrated.
End quote from Fath al-Baari (11/112).
One of the most useful and most well-known books concerning this matter is al-Adhkaar by Imam an-Nawawi (may Allah have mercy on him).
This is the way in which one may study the hadiths of the Prophet (blessings and peace of Allah be upon him).
And Allah knows best.