The istiadaha and basmalah are not part of the opening du`a reported from the Prophet (peace and blessings of Allah be upon him), although they are part of what should be recited at the beginning of the prayer. Aaishah (may Allah be pleased with her) said: The Messenger of Allah (peace and blessings of Allah be upon him) used to open his prayer with takbeer (Allahu akbar).
(Reported by Muslim, no. 498)
As regards saying the istiadhah and basmalah in the second rakah, the scholars differ on this matter. The basmalah should be recited in every rakah, when reciting al-Fatihah, but it should not be recited aloud, because there is some dispute as to whether it is part of al-Fatihah or not. Those scholars who say that it is not part of al-Fatihah recommend that it should be recited before al-Fatihah in each rakah.
So to be on the safe side, the worshipper should recite the basmalah, silently, in every rakah.
With regard to the istiadhah, there is no difference among the scholars, all of whom say that it is not part of Soorat al-Fatihah, but there is some discussion among them concerning the ayah (interpretation of the meaning): So when you want to recite the Quran, seek refuge with Allah [Al-Nahl 16:98]
Those who take the ayah at face value say that the istiadhah should be recited in every rakah, before reciting al-Fatihah and the basmalah. This is the opinion of the Shafais and is the most correct opinion in their view.
(Al-Majmoo, 3/323). Al-Hasan al-Basri, Ataa and Ibraheem al-Nakhai also recommended tawwudh in every rakah.
Ibn Hazm (may Allah have mercy on him) favoured this opinion, and it was also reported that Imam Muhammad ibn Abd al-Wahhab (may Allah have mercy on him) mentioned it in Aadab al-Mashiy ilal-Salah. This opinion was also favoured by Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him)
(al-Insaf 2/74), who said in al-Ikhtiyaaraat: Tawwudh is recommended at the beginning of every recitation. (p.50). This is also the opinion of al-Allamah al-Albani.
Some scholars say that it is to be recited at the beginning of the first rakah only, and does not have to be repeated in the other rakahs. This is the opinion of the Hanbalis
(al-Insaf, 2/73). Al-Shawkani thought that the correct opinion was to recite it in the first rakah only. (Nayl al-Awtar, 3/39, 139-140).
The Prophet (peace and blessings of Allah be upon him) used to seek refuge with Allah when he prayed. He used to say Aoodhu Billahi min al-shaytan il-rajeem min hamzihi wa nafkhihi wa nafathih (I seek refuge with Allah from the accursed Shaytan, from his evil insinuations).
(reported by Abu Dawood). Sometimes he would expand on this and say, Aoodhu Billahi il-Samee il-Aleem min al-shaytan il-rajeem min hamzihi wa nafkhihi wa nafathih (I seek refuge with Allah, the All-Hearing, the All-Seeing, from the accursed Shaytan, from his evil insinuations). (Reported by Abu Dawood and al-Tirmidhi). And Allah knows best.