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

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-Albaani 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-Bukhaari,
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‘aa’ 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: 

Allaahumma 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. Allaahumma aamantu bi kitaabika alladhi anzalta wa nabiyyika alladhi
arsalta (O Allaah 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 Allaah, 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 Allaah be upon him),
and when I reached the words. “Allaahumma aamantu bi kitaabika allaahi
anzalta (O Allaah, 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-Bukhaari (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.