How to offer Eid prayer:  
Ibn Abbass (ra) reported: " I participated in the Eid-ul-Fitr prayer with the Messenger of Allah (saw), Abu Bakr (ra), Umar (ra) and Uthman (ra), and all of them held Eid prayer before Khutbah, and then the Prophet Muhammad (saw) delivered the Khutbah (sermon)."  ( Muslim )

Who should go to the prayer ground & offer Eid Prayer:

Umm Atiyah (ra) reported: "The Messenger of Allah (saw) commanded us to bring out on Eid-al-Fitr and Eid-al-Adha, young women, hijab-observing adult women and the menstruating women. The menstruating women stayed out of actual Salaat but participated in good deeds and Duaa (supplication). I (Umm Atiyah) said to the Holy Prophet (saw): Oh! Messenger of Allah, one does not have an outer garment. He replied: Let her sister cover her with her garment."  ( Muslim )

On the Eid day, every believing man, woman and child must go to the prayer ground and participate in this joyous occasion.

Structure of Eid prayer:
Eid prayer is wajib (strongly recommended, just short of obligatory). It consists of two Rakaat (units) with six or thirteen additional Takbirs. It must be offered in congregation. The prayer is followed by the Khutbah.

The Khutbah is part of the worship and listening to it is Sunnah. During the Khutbah, the Imam must remind the community about its responsibilities and obligations towards Allah, fellow Muslims and the fellow human beings. The Imam must encourage the Muslims to do good and ward off evil. The Muslim community must also be directed to the state of the community and the Ummah at large and the feelings of sacrifice and struggle should be aroused in the community. At the conclusion of the prayer the Muslims should convey greetings to each other, give reasonable gifts to the youngsters and visit each other at their homes. Muslims should also take this opportunity to invite their non-Muslims neighbors, co-workers, classmates and business acquaintances to Eid festivities to expose them to Islam and Muslim culture.