Here are two authentic ḥadīth from our A’immah (عليهم السلام) where they have explicitly stated that the three days following Eid al-Fiṭr and Eid al-Aḍḥa, we should not fast, because those days are for eating and drinking.
We have ḥadīth about fasting six days in the month of Shawwāl, Hurr al-`Āmilī has interpreted and reconciled both of these narrations to mean that you can do the six days of fasting after the three days have passed.
ابْنُ أَبِي عُمَيْرٍ عَنْ زِيَادِ بْنِ أَبِي الْحَلَّالِ قَالَ قَالَ أَبُو عَبْدِ اللَّهِ ع لَا تَصُمْ بَعْدَ الْأَضْحَى ثَلَاثَةَ أَيَّامٍ وَ لَا بَعْدَ الْفِطْرِ ثَلَاثَةَ أَيَّامٍ إِنَّهَا أَيَّامُ أَكْلٍ وَ شُرْبٍ
From Ziyād b. Abī al-Ḥilāl said, Abū `Abd Allāh (عليه السلام) said, “Do not fast three days after (Eid) al-Aḍḥa, nor three days after (Eid) al-Fiṭr, those (days) are for eating and drinking”[1]
وَ الَّذِي رَوَاهُ عَلِيُّ بْنُ الْحَسَنِ بْنِ فَضَّالٍ عَنْ مُحَمَّدِ بْنِ إِسْمَاعِيلَ عَنْ حَمَّادِ بْنِ عِيسَى عَنْ حَرِيزٍ عَنْهُمْ ع قَالَ إِذَا أَفْطَرْتَ مِنْ رَمَضَانَ فَلَا تَصُومَنَّ بَعْدَ الْفِطْرِ تَطَوُّعاً إِلَّا بَعْدَ ثَلَاثٍ يَمْضِينَ
From Ḥarīz from them (عليه السلام) said, “When you do al-Fiṭr of (the month of) Ramaḍān, do not do voluntary fast after (Eid) al-Fiṭr, except after three (days) have passed”[2]
[1] al-Ṭūsī, Tahdhīb al-Aḥkām, vol. 4, ch. 72, pg. 330, ḥadīth # 99; al-Majlisī said this ḥadīth is Ṣaḥīḥ (Authentic) (See: al-Majlisī, Milādh al-Akhyār, vol. 7, pg. 170)
[2] al-Ṭūsī, Tahdhīb al-Aḥkām, vol. 4, ch. 67, pg. 298, ḥadīth # 5; al-Majlisī said this ḥadīth is Muwaththaq (Reliable) (See: al-Majlisī, Milādh al-Akhyār, vol. 7, pg. 113)
Bibliography
al-Majlisī. Milādh al-Akhyār. 1st. 16 vols. Qum: Maktabah Ayatollah al-Mar`ashī al-Najafī, 1406.
al-Ṭūsī. Tahdhīb al-Aḥkām. 4th. 10 vols. Tehran: Dār al-Kutub al-Islāmiyyah, 1407.