1) Performing the complete bathing (ghusl)
عن علي رضي الله عنه أنه سئل عن الغسل فقال: يوم الجمعه، ويوم العرفة، ويوم الفطر، ويوم الأضحى
'Ali رضي الله عنه was asked about the complete bathing and he replied, "(It is for) Friday (Jumu'ah), the Day of 'Arafah, the Day of Breaking the Fast ('Eid al-Fitr) and the Day of the Sacrifice ('Eid al-Ad-haa)."  2) Wearing one's best clothing
Ibn 'Abbaas رضي الله عنهما said,
كان رسول الله صلى الله عليه وسلم يلبس يوم العيد بردة حمراء
"The Messenger of Allaah صلى الله عليه وسلم used to wear a red cloak on the Day of 'Eid."  3) Eating on the Day of Breaking the Fast before going out for the prayer, and delaying eating on the Day of Sacrifice until one can eat from one's sacrificial animal
Anas رضي الله عنه said,
كان رسول الله صلى الله عليه وسلم لا يغدو يوم الفطر حتى يأكل تمرات
"The Messenger of Allaah صلى الله عليه وسلم would not depart on the Day of Breaking the Fast until he had eaten some dates." 
Abu Buraydah رضي الله عنه said,
أن رسول الله صلى الله عليه وسلم كان لا يخرج يوم الفطر حتى يطعم ، ولا يطعم يوم النحر حتى يذبح
"The Messenger of Allaah صلى الله عليه وسلم would not depart on the Day of Breaking the Fast until he had eaten. And he would not eat on the Day of Sacrifice until he had slaughtered (the sacrificial animal)."  4) Taking a different path while returning from the prayer
Jaabir رضي الله عنه said,
كان النبي صلى الله عليه وسلم إذا كان يوم عيد خالف الطريق
"On the Day of 'Eid, the Prophet صلى الله عليه وسلم would take a different path (when returning from the prayer)."  5) The saying of the Takbeer during the days of 'Eid
Allaah سبحانه و تعالى has said: "The month of Ramadaan in which was revealed the Qur'aan, a guidance for mankind and clear proofs for the guidance and the criterion (between right and wrong). So whoever of you sights (the crescent on the first night of) the month (of Ramadaan i.e. is present at his home), he must observe Saum (fasts) that month, and whoever is ill or on a journey, the same number [of days which one did not observe Saum (fasts) must be made up] from other days. Allaah intends for you ease, and He does not want to make things difficult for you. (He wants that you) must complete the same number (of days), and that you must magnify Allaah [i.e. to say Takbeer (Allaahu-Akbar; Allaah is the Most Great) on seeing the crescent of the months of Ramadaan and Shawwaal] for having guided you so that you may be grateful to Him."
This is concerning the 'Eid of Breaking the Fast . About the 'Eid of Sacrifice, Allaah سبحانه و تعالى says: "And remember Allaah during the appointed Days. But whosoever hastens to leave in two days, there is no sin on him and whosoever stays on, there is no sin on him, if his aim is to do good and obey Allaah (fear Him), and know that you will surely be gathered unto Him."
And also: "It is neither their meat nor their blood that reaches Allaah, but it is piety from you that reaches Him. Thus have We made them subject to you that you may magnify Allaah for His Guidance to you. And give glad tidings (O Muhammad صلى الله عليه وسلم) to the Muhsineen (doers of good)."
On the Day of Breaking the Fast, the takbeer is to be said from the time the person leaves to go to the place of prayer until the time that he prays. Ibn Abi Shaybah رحمه الله reported that Yazeed ibn Haaroon رحمه الله had informed him on the authority of Ibn Abi Dhi'b رحمه الله that az-Zuhri رحمه الله said,
أن رسول الله صلى الله عليه وسلم كان يخرج يوم الفطر فيكبر حتى يأتى المصلى، وحتى يقضى الصلاة، فإذا قضى الصلاة قطع التكبير
"The Messenger of Allaah صلى الله عليه وسلم would go out on the Day of Breaking the Fast and he would make the takbeer until he reached the place of the prayer and until he finished his prayer. Once he finished the prayer, he discontinued the takbeer." 
The timing for the takbeer during the Days of the Sacrifice is from the morning of the Day of 'Arafah until the afternoon of the last day of the Days of Tashreeq. That has been authentically narrated from 'Ali, Ibn 'Abbaas and Ibn Mas'ood (may Allaah سبحانه و تعالى be pleased with them). 
As for what one should say in the takbeer, the matter is flexible. It is confirmed from lbn Mas'ood رضي الله عنه that he would make the statement of takbeer twice, saying during the days of Tashreeq:
الله اكبر ، الله أكبر ، لا اله إلا الله ، الله أكبر ، الله أكبر ولله الحمد
Allaahu akbar, Allaahu akbar, laa ilaaha illallaah, Allaahu akbar, Allaahu akbar, wa lillaahil-hamd
"Allaah is the Greatest, Allaah is the Greatest, there is none worthy of worship except Allaah, Allaah is the Greatest, Allaah is the Greatest, and to Allaah belongs the praise."   Notes:
 This hadeeth is saheeh. See Shaykh al-Albaani رحمه الله, Irwaa' al-Ghaleel , vol. 1, p. 177. Recorded by al-Bayhaqi.
 The chain of this report is good (jayyid). See Shaykh al-Albaani رحمه الله, Silsilat al-Ahaadeeth as-Saheehah, no. 1279. Imaam al-Haythami رحمه الله stated in Majma' az-Zawaa'id, vol. 2, p. 201, "It is recorded by at-Tabaraani in al-Awsat and its narrators are trustworthy."
 This hadeeth is saheeh. See Shaykh al-Albaani رحمه الله, Saheeh Sunan at-Tirmidhi, no. 448. Recorded by al-Bukhaari and at-Tirmidhi.
 This hadeeth is saheeh. See Shaykh al-Albaani رحمه الله, Saheeh Sunan at-Tirmidhi, no. 447. Recorded by Ibn Khuzaymah and by at-Tirmidhi but with the words, "until he prayed."
 This hadeeth is saheeh. See Shaykh al-Albaani رحمه الله, Mishkaat al-Masaabeeh, no. 1343. Recorded by al-Bukhaari.
 This hadeeth is saheeh. See Shaykh al-Albaani رحمه الله, Silsilat al-Ahaadeeth as-Saheehah, no. 171. Shaykh al-Albaani رحمه الله said, "This chain is saheeh, missing the name of the Companion only. It is narrated through another way on the authority of 'Umar from the Prophet صلى الله عليه وسلم. Al-Bayhaqi recorded from the chain of 'Abdullaah ibn 'Umar from Naafi' رحمه الله from 'Abdullaah ibn 'Umar, رضي الله عنهما,
أن رسول الله صلى الله عليه وسلم كا يخرج في العيدين مع الفضل بن عباس وعبد الله بن عباس وعلي وجعفر والحسن والحسين وأسامة بن زيد وزيد بن حارثة وأيمن بن أم أيمن رضي الله عنهم رافعاً صوته بالتحليل والتكبير، فيأخذ طريق الحذائين حتى يأتى المصلى، فإذا فرغ رجع على الحذاءين حتى يأتى منزله
'The Messenger of Allah صلى الله عليه وسلم used to go out for the two 'Eids with al-Fadl ibn 'Abbaas, 'Abdullaah ibn 'Abbaas, 'Ali, Ja'far, al-Hasan, al-Husayn, Usaamah ibn Zayd, Zayd ibn Haarithah and Ayman ibn Umm Ayman (may Allaah سبحانه و تعالى be pleased with them). He would be raising his voice saying, "There is none worthy of worship except Allaah' and 'Allaah is greatest."'
Imaam al-Bayhaqi رحمه الله said, 'This chain is better than the previous one.' I (that is, Shaykh al-Albaani رحمه الله) say, 'Its narrators are trustworthy and are the narrators of Imaam Muslim رحمه الله, save for 'Abdullaah ibn 'Umar رحمه الله who is al-'Umari al-Mukabbir.' Imaam adh-Dhahabi رحمه الله said about him, 'He is honest but there is some problem with his memory.' He and others put the notation that he is from the narrators of Imaam Muslim رحمه الله. Someone like that can be used as supporting evidence. Thus, this is a sound supporting evidence for the report of Imaam az-Zuhri رحمه الله which is missing the name of the Companion only. Therefore, the hadeeth is authentic, in my opinion, both as a statement of a Companion and as a statement of the Prophet صلى الله عليه وسلم. And Allaah سبحانه و تعالى alone knows best." Irwaa' al-Ghaleel, vol. 3, p. 123.
 Ibn Abi Shaybah (vol. 2, p. 165) narrated it from 'Ali رضي الله عنه via two chains, one of them good. And from this means it was recorded by al-Bayhaqi (vol. 3, p. 314). Then he recorded similar from Ibn 'Abbaas رضي الله عنهما with a sound chain. Al-Haakim recorded it from him and from Ibn Mas'ood رضي الله عنه similarly. See Shaykh al-Albaani رحمه الله, Irwaa' al-Ghaleel, vol. 3, p. 125.
 Ibn Abi Shaybah recorded this with a saheeh chain. However, as another location, with the same chain, he mentioned that he said the takbeer three times. This is how al-Bayhaqi records it from Yahyaa ibn Sa'eed رحمه الله from al-Haakim رحمه الله, who is Ibn Farooh Abu Bakaar رحمه الله, from 'Ikrimah رحمه الله, from Ibn 'Abbaas رضي الله عنهما, saying the takbeer three times. And the chain of that report is also saheeh. See Shaykh al-Albaani رحمه الله, Irwaa' al-Ghaleel, vol. 3, p. 125.
 Shaykh Dr. Abdul-Azeem Badawi حفظه الله, The Concise Presentation of the Fiqh of the Sunnah and the Noble Book, pp. 212-216.