What should be learned of the Qur’aan and hadeeth?
Question

                Which are the quran surah and ayats that every single muslim should memorize them.and what the sunnah which every muslim should know and perform them.
Praise be to Allah.Some Islamic knowledge is 
    obligatory for every accountable individual to learn. This is called fard 
    ‘ayn (individual obligation). The definition is: if it has to do with an act 
    of worship that he is obliged to do, or a transaction that he wants to 
    engage in (then he must learn it). 
And some knowledge is 
    obligatory for the ummah in general: if some of them do it in the required 
    manner then the rest are absolved of responsibility, otherwise all of those 
    who are able to do it are sinning. This is called fard kifaayah (communal 
    obligation). 
Memorizing Qur’aan and 
    hadeeth comes under the leading of the latter. The ummah is obliged to 
    preserve the Book of Allaah and the hadeeth of His Messenger, whether in 
    their hearts or on paper, so that nothing of the two sources of legislation 
    will be lost. Imam al-Nawawi (may Allaah have mercy on him) said in the 
    introduction to Sharh al-Muhadhdhab (1/51): The second type – i.e., 
    of Islamic knowledge – is a communal obligation (fard kifaayah), which means 
    acquiring that Islamic knowledge which is essential for the people to 
    establish their religion, such as memorization of Qur’aan and hadeeth, and 
    their sciences, and usool al-fiqh.  
As for memorization of 
    Qur’aan and hadeeth as an individual obligation – i.e., that which the 
    accountable Muslim is required to memorize – he is not obliged to memorize 
    any of that by heart, he only has to memorize that without which prayer is 
    not valid. Al-Faatihah has to be learned by every accountable person, i.e., 
    he has to learn how to recite it properly. If he learns it by heart that is 
    good; if he does not learn it by heart then he must read it from the Mus-haf 
    or a piece of paper and the like. 
Al-Nawawi said in 
    al-Majmoo’ (3/335): Our companions said: If a person is not able to 
    recite al-Faatihah then he should acquire the ability to do so by learning 
    it, or acquire a Mus-haf from which he can read it, by buying it or renting 
    it or borrowing it. If he is praying at night or in the dark, then he must 
    acquire a lamp if he can. If he refuses to do that when he is able to do it, 
    then he is sinning and must repeat every prayer that he offered before he 
    learned al-Faatihah. Our evidence is the well-known shar’i principle that 
    everything without which an obligatory duty cannot be performed, when it is 
    within a person’s capabilities, is obligatory. End quote. 
In Kashshaaf al-Qinaa’ 
    (1/34) by al-Bahooti (may Allaah have mercy on him) it says: “The ignorant 
    person” i.e., the one who cannot recite al-Faatihah well “must learn it” 
    because it is obligatory in prayer, so he has to learn it if he is able to, 
    as a condition (of prayer). “If he does not do that” i.e., if he does not 
    learn al-Faatihah, “when he is able to do it, then his prayer is not valid” 
    because he has failed to do an obligatory action when he was able to do it. 
    End quote. 
Mustafa al-Ruhaybaani 
    al-Hanbali said in Mataalib Ooli al-Nuha (1/432): The ignorant person 
    must learn al-Faatihah because it is obligatory in prayer, so he must learn 
    it if he is able, as it is a condition of prayer. End quote. 
The same applies to 
    hadeeth: Muslim men and women are not required to memorize any of them, but 
    it is essential to learn that without which the prayer is invalid, such as 
    the hadeeth which describes the wording of the tashahhud to be recited in 
    prayer. Ibn Qudaamah (may Allaah have mercy on him) said in al-Mughni 
    (1/315): The minimum that is acceptable is Al-Tahiyyaatu Lillaahi 
    wa’l-salaamu ‘alayka ayyhuha’l-nabiyyu wa rahmat-Allaah, al-salaamu ‘alayna 
    wa ‘ala ‘ibaad Illaah il-saaliheen, ashhadu an la ilaaha ill-Allaah wa 
    ashhadu anna Muhammadan ‘abduhu wa rasooluhu (or anna Muhammadan 
    rasool-Allaah) [All compliments are due to Allaah, Peace be upon you, O 
    Prophet, and the mercy of Allaah, Peace be upon us, and on the righteous 
    slaves of Allaah. I bear witness that there is no god except Allaah, and I 
    bear witness that Muhammad is His slave and Messenger (or that Muhammad is 
    the Messenger of Allaah)].”
End quote. 
And Allaah knows best.