Yes, there are special Sunnahs for Eid day. Amongst the Sunnahs of Eid day are, eating before leaving for the Eid prayer, making Ghusl, using Siwak, using scent (Atr) and many others.
عَنْ أَنَسِ بْنِ مَالِكٍ، قَالَ: كَانَ رَسُولُ اللَّهِ صَلَّى اللهُ عَلَيْهِ وَسَلَّمَ لاَ يَغْدُو يَوْمَ الفِطْرِ حَتَّى يَأْكُلَ تَمَرَاتٍ
(Sayidina) Anas bin Malik (May Allah Most High be pleased with him) said, ‘The Messenger of Allah ﷺ would not go out on the day of Eid al-Fitr until he ate some dates.’ (Sahih al-Bukhari, 953).
The Hadith above mentions one of numerous Sunnahs narrated for Eid day. The Hanafi scholars have mentioned them in the books of Fiqh:
وندب في الفطر أن يطعم ويغتسل ويستاك ويتطيّب ويلبس أحسن ثيابه ويؤدّي صدقة الفطر ثمّ يتوجّه إلى المصلّى. (كنز الدقائق)
And it is recommended for (Eid) al-Fitr to eat, perform Ghusl, use Siwak, use scent (Atr), wear one’s best clothes, pay Sadaqatu al-Fitr and then head to the place of prayer. (Kanz al-Daqaiq, Imam Abu al-Barakaat al-Nasafi).
The list provided above by Imam al-Nasafi should not be considered exhaustive. However, it does mention the main Sunnahs for Eid al-Fitr.
There is a little difference between the Sunnahs of Eid al Fitr and Eid al-Adha. Although most of the above are Sunnah for both, the recommendation of eating before going for Eid Salah is recommended for Eid al-Fitr but not for Eid al-Adhaa.
Eid is also a great opportunity to spend time with one’s family. It has become an occasion to connect with family members whom a person sees infrequently. Therefore, one should endeavour to participate in this and meet as many family members as possible on the day of Eid. If meeting is not possible then at the very least send them an Eid greetings message.
It is also important for the Muslim community to keep new Muslims in mind. During this period they can feel lonely or excluded. Since the community is all gathering with family, and their family may not celebrate Eid. Therefore, every community should make sure something is organised to support the new Muslims in the community.
And Allah Most High Knows Best
-Answered by Shaykh Noorud-deen Rashid (30.04.2022)
Is Eid prayer Fard?
See also (video):