… gleaned from Baha’i literature

From Childhood Baha’u’llah was “gifted in every way, and adorned with purity and nobility”

…There was in Ṭihrán… a Youth of the family of one of the ministers and of noble lineage, gifted in every way, and adorned with purity and nobility. Although He combined lofty lineage with high connection, and although His ancestors were men of note in Persia and universally sought after, yet He was not of a race of doctors or a family of scholars. Now this Youth was from His earliest adolescence celebrated amongst those of the ministerial class, both relatives and strangers, for single-mindedness, and was from childhood pointed out as remarkable for sagacity, and held in regard in the eyes of the wise. He did not, however, after the fashion of His ancestors, desire elevation to lofty ranks nor seek advancement to splendid but transient positions. His extreme aptitude was nevertheless admitted by all, and His excessive acuteness and intelligence were universally avowed. In the eyes of the common folk He enjoyed a wonderful esteem, and in all gatherings and assemblies He had a marvelous speech and delivery. Notwithstanding lack of instruction and education such was the keenness of His penetration and the readiness of His apprehension that when during His youthful prime He appeared in assemblies where questions of divinity and points of metaphysic were being discussed, and, in presence of a great concourse of doctors and scholars loosed His tongue, all those present were amazed, accounting this as a sort of prodigy beyond the discernment natural to the human race. From His early years He was the hope of His kindred and the unique one of His family and race, nay, their refuge and shelter.
- ‘Abdu’l-Baha  (‘A Traveler’s Narrative’)

Bahá'u'lláh became renowned for His learning when He was about thirteen years old

He was extremely kind and generous. He was a great lover of outdoor life, most of His time being spent in the garden or the fields. He had an extraordinary power of attraction, which was felt by all. People always crowded around Him, Ministers and people of the Court would surround Him, and the children also were devoted to Him. When He was only thirteen or fourteen years old He became renowned for His learning. He would converse on any subject and solve any problem presented to Him. In large gatherings He would discuss matters with the 'Ulama (leading mullas) and would explain intricate religious questions. All of them used to listen to Him with the greatest interest.
- 'Abdu'l-Baha  (Quoted by Esslemont in ‘Baha'u'llah and New Era’)

Bahá'u'lláh didn’t attend any schools

The education and instruction which Bahá'u'lláh received was limited both in nature and extent, as He Himself states in the Tablet addressed to Nasir’d-Din Shah: 'The learning current amongst men I studied not; their schools I entered not. Ask of the city wherein I dwelt, that thou mayest be well assured that I am not of them who speak falsely.'

In those days, the scions of noble houses were taught such matters as befitted their station in life, such as riding, handling a gun, wielding a sword, calligraphy, acquaintance with the works of the great classical poets of the land, a good reading knowledge of the Holy Book, the Qur'an, and hardly ever anything more. They were given such instruction by tutors, specially engaged by the parents, who were also required to teach them good manners.
- Balyuzi  (‘Bahá'u'lláh, The King of Glory’)

Bahá'u'lláh’s father had an amazing dream about his Son

Baha'u'llah's father
When Bahá'u'lláh was still a child, the Vazir, His father, dreamed a dream. Bahá'u'lláh appeared to him swimming in a vast, limitless ocean. His body shone upon the waters with a radiance that illumined the sea. Around His head, which could distinctly be seen above the waters, there radiated, in all directions, His long, jet-black locks, floating in great profusion above the waves. As he dreamed, a multitude of fishes gathered round Him, each holding fast to the extremity of one hair. Fascinated by the effulgence of His face, they followed Him in whatever direction He swam. Great as was their number, and however firmly they clung to His locks, not one single hair seemed to have been detached from His head, nor did the least injury affect His person. Free and unrestrained, He moved above the waters and they all followed Him.

The Vazir, greatly impressed by this dream, summoned a soothsayer, who had achieved fame in that region, and asked him to interpret it for him. This man, as if inspired by a premonition of the future glory of Bahá'u'lláh, declared: "The limitless ocean that you have seen in your dream, O Vazir, is none other than the world of being. Single-handed and alone, your son will achieve supreme ascendancy over it. Wherever He may please, He will proceed unhindered. No one will resist His march, no one will hinder His progress. The multitude of fishes signifies the turmoil which He will arouse amidst the peoples and kindreds of the earth. Around Him will they gather, and to Him will they cling. Assured of the unfailing protection of the Almighty, this tumult will never harm His person, nor will His loneliness upon the sea of life endanger His safety."

As a boy Baha’u’llah was amazed at pointless discourses by some clerics

Baha'u'llah ... records that on one occasion, when visiting his future mother-in-law, he listened to a cleric with a large turban expounding on whether the arch-angel Gabriel was the greater or Qanbar, the servant of the Imam ‘Ali. He writes that although he was still but a boy, he was astonished at the ignorance of these elders and when an opportunity came, he expressed his thoughts saying that since Gabriel is stated in Qur’an to be the one through whom the Word of God is revealed to the prophet Muhammad, then even Qanbar's master the Imam ‘Ali would not reach that station. Baha'u'llah states that some time later, he visited Qum and was dismayed to find the same cleric expounding the same sort of pointless and fruitless discourse.
- Moojan Momen  ('Baha’u’llah, A Short Biography')