Praise be to Allah.
The Sunnah for the one who comes to the Eid prayer-place to pray the Eid prayers or to offer prayers for rain, is to sit down and not pray Tahiyyat al-Masjid (two rak’ahs to greet the mosque), because it is not narrated that the Prophet (peace and blessings of Allaah be upon him) or his companions did that, as far as we know. But if the prayer is being held in a mosque, then he should pray Tahiyyat al-Masjid, because of the general applicability of the hadeeth of the Prophet (peace and blessings of Allaah be upon him), “When any one of you enters the mosque, let him not sit down until he has prayed two rak’ahs.” (Saheeh, agreed upon).
What is prescribed for the one who sits down to await the Eid prayer is to recite a lot of Tahleel (saying Laa ilaaha ill-Allaah) and Takbeer (saying Allaahu akbar), because these are the slogans of this day, and this is the Sunnah for all of those who are inside the mosque or outside of it, until the khutbah is over. If a person busies himself with reading Qur’aan, this is also OK. And Allaah is the Source of strength.