Should I Recite “A’oodhu Billah” in every Rak`ah in Prayer?
Question

Must a person say ‘A’oodhu Billahi min al-Shaytan ir-rajeem’ (I seek refuge with Allah from the accursed Shaytan) in each rak`ah or only in the first?
Praise be to Allah.Isti`adhah (seeking refuge with Allah)
Allah has commanded us to seek refuge with Him from the accursed Shaytan when reciting Quran. Allah says (interpretation of the meaning): 
“So when you want to recite the Quran, seek refuge with Allah from Shaytan (Satan), the outcast (the cursed one).” [al-Nahl 16:98]
The meaning of ‘A’oodhu Billahi min al-Shaytaan ir-rajeem’
What seeking refuge with Allah means is turning to Allah for protection from the evil of every evil one. It is said in order to ward off evil from a person. It is as if the one who says it is saying: I am seeking refuge with Allah from the accursed Shaytan lest he harm me in my spiritual or worldly affairs, or he prevents me from doing what I have been commanded or he makes me do something that I have been forbidden to do. 
Is it obligatory to say ‘A’oodhu Billahi min al-Shaytan ir-rajeem’ ?
The majority of scholars say that saying “A’oodhu Billah…” is mustahabb (encouraged), it is not obligatory. 
Some scholars said that it is obligatory, quoting this verse as evidence and saying that the command mentioned therein implies that it is obligatory. This is the view of Ibn Hazm, and Ibn Katheer was also inclined towards this view. (See Tafseer Ibn Katheer, 1/14)
The Standing Committee favoured the view that saying “A’oodhu Billah…” is Sunnah. 
It says in Fatawa al-Lajnah al-Daimah: 
“Saying “A’oodhu Billah…” is Sunnah and if a person does not say it in his prayer, either deliberately or because he forgot, that does not affect him.” (Fatawa al-Lajnah al-Daimah, 6/381)
Formulas of saying ‘A’oodhu Billahi min al-Shaytan ir-rajeem’ 
There are different ways of saying these words of seeking refuge with Allah: 
1-A’oodhu Billahi min al-Shaytan ir-rajeem (I seek refuge with Allah from the accursed Shaytan)
2-A’oodhu Billah as-samee’ il-‘aleem min al-Shaytan ir-rajeem (I seek refuge with Allah, the All-Hearing, All-Knowing, from the accursed Shaytan)
3-Adding to the second version the words min hamzihi wa nafkhihi wa nafathihi (from his suffocation, his arrogance and his poetry). (See Tafseer Ibn Katheer, 1/13)
Should we recite  ‘A’oodhu Billahi min al-Shaytan ir-rajeem’ in every rak`ah?
The words “A’oodhu Billah…” may be said during prayer and at other times. In prayer it is sufficient to recite them once, in the first rak’ah. This does not have to be said in every rak’ah. 
Ibn Qudamah said: 
“Muslim narrated that Abu Hurayrah said: “When the Messenger of Allah (peace and blessings of Allah be upon him) got up from the second rak’ah, he would start to recite al-Fatihah from the words Al-hamdu Lillahi Rabb il-‘Alameen, and he did not pause beforehand.” This indicates that he did not recite the opening du’a or say “A’oodhu Billah…” 
With regard to saying “A’oodhu Billah…” there is a different report from Ahmad which says that it should be said in every rak’ah. It was also narrated from him that it should be said in the first rak’ah only. This is the view of ‘Ata, al-Hasan, al-Nakha’i and al-Thawri, because of the hadeeth of Abu Hurayrah mentioned above, and because the prayer is considered to be a single action, so it does not have to be said again. 
The second report, which is that one should say “a’oodhu Billah…” in each rak’ah, is the view of Ibn Sireen and al-Shafi'i, because Allah says (interpretation of the meaning):
“So when you want to recite the Quran, seek refuge with Allah from Shaytan (Satan), the outcast (the cursed one)” [al-Nahl 16:98]
That implies that one should repeat the words “a’oodhu Billah…” each time one wants to recite again, because it is prescribed when reciting Quran, so if you want to recite Quran again you should say these words again, as if each rak’ah is an independent action.” (Al-Mughni, 2/216)
Ibn al-Qayyim (may Allah have mercy on him) said: 
“The view that it is sufficient to say “a’oodhu Billah…” once is more correct.” (Zad al-Ma'ad, 1/242)
Al-Shawkani said: 
“The ahadeeth which were narrated concerning seeking refuge with Allah indicate only that this should be done in the first rak’ah… 
In order to be on the safe side, we should limit ourselves to what has been narrated in the Sunnah, which is to say “a’oodhu Billah…” before the first rak’ah only.” (Nayl al-Awtar, 2/231)
When should we recite A’oodhu Billahi min al-Shaytan ir-rajeem’ in prayer?
The fuqaha differed as to where in the prayer the words “a’oodhu Billah…” should be said. Some said that it should be after reciting Quran, which is a da’eef (weak) view. 
Ibn Katheer (may Allah have mercy on him) said: 
‘The well-known view, which is that of the majority, is that the words “a’oodhu Billah…” should be said before reciting Quran, to ward off whispers from the Shaytan therein.” (Tafseer al-Quran il-‘Azeem, 1/13)
Al-Jassas said: 
“The view of those who say that the words “a’oodhu Billah…” should be said after finishing the recitation from the Quran is odd. Rather these words should be said before reciting, to ward off the whispers of the Shaytan from the recitation. Allah says (interpretation of the meaning): 
“Never did We send a Messenger or a Prophet before you but when he did recite the Revelation or narrated or spoke, Shaytaan (Satan) threw (some falsehood) in it. But Allah abolishes that which Shaytaan (Satan) throws in” [al-Hajj 22:52] 
Allah has commanded us to seek refuge with Him before reciting, for this reason.” (Ahkaam al-Quran, 3/283)
For other issues related to adhkar, please see these answers: 112098 , 105396 , and 147123 .
And Allah knows best.