The Eid Prayer (Salat al-Eid) consists of two Rak`ahs.
(Ayt Sistani method) In the first Rak`ah, recite Surah of al-Fatihah and Surah Al A'ala ( or any other) After the recitation of these Surahs, say the takbir 'Allahu-akbar' - four times and raise the hands for the Qunut after each takbeer, saying as below ( or any other qunoot): then say the takbeer for fifth time after 4th qunoot and again for the sixth time before ruku , then do the ruku and sujud.
In the second Rak`ah, recite Surah of al-Fatihah and Surah 91 As Shams ( or any other)and then say the takbeer three times raising the hands for the Qunut after each takbeer After that, say the fourth takbeer after 3rd Qunoot & fifth takbeer before ruku and continue to complete the prayer.
Also See Notes below
Qunoot in Eid prayer ( 4 times in first rakat & 3 times in 2nd Rakat)

اللّهُمّ أَهْلَ الْكِبْرِيَاءِ وَالْعَظَمَةِ،

allahumma ahla alkibriya‘i wal-`azamati

O Allah: You are the Lord of glory and greatness

Qunoot Dua :

وَأَهْلَ الْجُودِ وَالْجَبَرُوتِ،

wa ahla aljudi wal-jabaruti

And the Lord of magnanimity and omnipotence

وَأَهْلَ الْعَفْوِ وَالرّحْمَةِ،

wa ahla al-`afwi wal-rrahmati

And the Lord of pardon and mercy,

وَأَهْلَ التّقْوَى وَالْمَغْفِرَةِ،

wa ahla alttaqwa wal-maghfirati

And the worthiest of being feared and the Lord of forgiveness;

أَسْأَلُكَ بِحَقّ هذَا الْيَوْمِ الّذِي جَعَلْتَهُ لِلْمُسْلِمِينَ عِيداً،

as’aluka bihaqqi hadha aliyawmi alladhy ja`altahu lilmuslimena `edan

I beseech You in the name of this day, which You have decided to be feast for the Muslims,

وَلِمُحَمّدٍ صَلَّى اللهُ عَلَيْهِ وَآلِه ذُخْراً وَمَزِيداً

wa limuhammadin salla allahu `alayhi wa alihi dhukhran wa mazedan

And to be safety and increasing honor for Muhammad—peace of Allah be upon him and his Household—

أَنْ تُصَلّيَ عَلَى مُحَمّدٍ وَآلِ مُحَمّدٍ

an tusalliya `ala muhammadin wa ali muhammadin

(I beseech You) that You may send blessings upon Muhammad and the Household of Muhammad,

وَأَنْ تُدْخِلَنِي فِي كُلّ خَيْرٍ أَدْخَلْتَ فِيهِ مُحَمّداً وَآلَ مُحَمّدٍ،

wa an tudkhilany fe kulli khayrin adkhalta fehi muhammadan wa ala muhammadin

And You may include me with every item of goodness with which You have included Muhammad and the Household of Muhammad,

وَأَنْ تُخْرِجَنِي مِنْ كُلِّ سُوءٍ أَخْرَجْتَ مِنْهُ مُحَمّداً وَآلَ مُحَمّدٍ

wa an tukhrijany min kulli su‘in akhrajta minhu muhammadan wa ala muhammadin

And that You save me from any item of evil from which You have saved Muhammad and the Household of Muhammad,

صَلَوَاتُكَ عَلَيْهِ وَعَلَيْهِمْ.

salawatuka `alayhi wa `alayhim

May Your blessings be upon him and upon them.

اللّهُمّ إِنّي أَسْأَلُكَ خَيْرَ مَا سَأَلَكَ عِبَادُكَ الصَّالِحُونَ،

allahumma inne as’aluka khayra ma sa’alaka `ibaduka alssalihuna

O Allah: I pray You for the best of all that for which Your righteous servants have prayed You,

وَأَعُوذُ بِكَ مِمَّا اسْتَعَاذَ مِنْهُ عِبَادُكَ الصَّالِحُونَ.

wa a`udhu bika mimma ista`adha minhu `ibaduka alssalihuna

And I seek Your protection against all that against which Your righteous servants have asked Your protection.
After completion of the prayer, say the famous Tasbeh al-Zahra' (a.s).