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After Lord Krishna kills evil king Kansa and frees his parents from prison, he studies at an Ashram.
The teacher says he is pleased with Lord Krishna's progress because he learned 64 Arts and 14 Sciences in 64 days.
What are these 64 Arts and 14 Sciences. Here is the video clip
The sixty four arts and fourteen sciences were taught in the past in the Gurukulas. The names of these arts and sciences along with a brief description are present in Shukra Niti.
Shukracharya in the Shukra Niti (moral code of conduct) says who is an expert of arts and sciences should be a preceptor. Who is unlearned should not be a preceptor. This text mainly discusses duties of princes, Kings, war and political strategies to be followed by a king to rule the kingdom prosperously.
Having saluted and duly worshipped the Stay of the Uni- verse, the Cause of the origin, maintenance and destruction (of the Creation), Shukracharya, the offspring of Bhrigu, questioned with respectful decorum by Purvadevas or Asuras, his disciples, gave them a discourse on the essence of morals in the logical order. [Chapter 1 Verse 1 of Shukra Niti]
From Chapter IV section III of Shukra Niti,
The arts and sciences are infinite and cannot be enumerated.
The primary Vidyas are thirty. The primary arts are sixty four.
Vidya is one which can be said. Kala is that which can be acted even by a dumb.
51-52. Rig, Yajus, Sama, Atharvan are the Vedas; Ayus, Dhanus,Gandharva, as well as Tantras are the Upavedas.
53-54. The six Añgas of the Vedas are Sikshâ (Pronunciation), Vyâkar ana. (Grammar), Kalpa (Rituals), Nirukta (Etymology), Jyoti (Astronomy) and Chhandas (Prosody).
In general, the above i.e., 4 vedas, 4 Upavedas and 6 Vedangas are collectively known as 14 sciences.
55-59. The Mimamsâs, Tarka, Samkhya, Vedanta, Yoga, Itihasas, Puranas, Smritis, theory of Sceptics, Artha Sâstra, Kama shastra, Silpa Sstra, Alañkâra (Rhetoric), Kavyas, language of the folk (vernacular), the art of spenking properly, the theory of Yavanas, and manners and customs of countries and nations—these are the thirty-two Vidyâs.
The text gives a description of all the Vidyas said above in a line or two.
The species or Kala is named after the function it serves (the work it does).
Nartan or dancing with appropriate gestures and movements is an art.
vAdana or playing on musical instruments is also an art.
The decoration of men and women by dress and ornaments is an art.
The performance and knowledge of t.he sundry mimicry and antics is an art.
The laying out of beds and furniture and the weaving of garlands, &c., constitute an art.
138-139. The entertainment of people by gambling and various tricks of magic is an art. The (knowledge of) different aspects of giving pleasure is an art.
These seven arts are called Gāndhara.
The distillation of wines and spirituous liquors from flowers, &c., is an art.
The extrication of thorns and the relieving of pain by operating on the wounds of a vein constitute an art.
The cooking of food by intermixtures of various tastes is an art.
The planting, grafting and preservation of plants constitute an art.
The melting and powdering of stones and metals constitute an art.
The act of using preparations from sugarcanes is known to be an art.
The knowledge of mixtures of metals and medicinal plants Constitutes an art.
The knowledge;of the analysis and synthesis of metals constitute an art.
The preparation of new substances (alloys) out of metals by combinations is an art.
The preparation of salts constitutes an art.
These ten Kalās are mentioned in Ayurveda and other (medical) sciences.
The use and employment of arms by the proper arrangement of legs constitutes an art.
Duelling by the various artifices is an art.
A Bāhuyuddha or hand to hand fight is the combat between duellers without weapons.
156-157. The Niyuddha is meant for fame only, destruction of the enemy’s power and vanity. A hand to hand fight should not lead to anybody’s death.
158-60. An attack by dwellers, that which is made by various dangerous artifices of hands, and by throwing down the opponent in various ways, ‘&c. And Pratikriya is the method of extricating oneself from these.
The throwing of arms and implements towards some fixed point is an art.
The formations of battle arrays according to the signals given by musical instruments (bugles) s an art.
The arrangement of horses, elephants and chariots in war is an art.
These five arts are mentioned in Dhanurveda or the science of military tactics.
The propitiation of gods by various seats and postures is an art.
166 The act of driving horses and elephants is an art, as well as that of teaching them.
167-168. Earthen, wooden, stone and metal vessels give rise to four separate arts in the matter of their cleansing, polishing, dyeing or rinsing; picture-drawing is also an art.
The construction of tanks, canals, palaces, and squares (?) is an art.
The construction of clocks, watches and musical instruments is an art.
The act of putting down the actions of water, air and re is an art.
The preparation of boats, chariots and conveyances is an art.
The preparation of threads and ropes is an art.
The weaving of fabrics by various threads is an art.
The testing of gems s to whether they are good or bad as possessing marks of holes is an art.
The testing of gold and other metals is an art.
The preparation of artificial gold and gems is an art.
The making of ornaments with gold and other metals is an art, as well as enamelling of metals.
The softening of leathers is an art.
The flaying of skins from the bodies of the beast is an art.
Milking and churning constitute two arts.
The knowledge of sewing of covers (coats and shirts) is an art.
Crossing waters by arms (swimming is an art.)
The cleansing of domestic utensils in an art.
Cleaning of clothes and shaving are two arts.
The extraction of oil from seeds and flesh (fats) is an art.
The drawing of plouglis and the climbing of trees are two arts.
The knowledge of work in such a way aS to please somebody is an art.
The making of vessels with bamboo straws, etc., is an art.
The making of glass vessels is an art.
The pumping and withdrawing of water constitute an art.
The preparation of tools and implements from iron is an art.
The preparation of saddles for horses, elephants, bulls and camels is an art.
The maintenance, and entertainment, and nursing of children constitute an art.
The punishment of offenders e.g., whipping, is an art.
The writing of the characters of various languages is an art.
The making and preservation of the betels constitute an art.
199-200. Speed in taking, but delay in giving—these are the two features of all arts.
Verses are taken from English Translation of THE SACRED BOOKS OF THE HINDUS translated by various authors.
According to Bhagavatam, Balarama and Krishna learnt each art in a day.
O King, those best of persons, Kṛṣṇa and Balarāma, being Themselves the original promulgators of all varieties of knowledge, could immediately assimilate each and every subject after hearing it explained just once. Thus with fixed concentration They learned the sixty-four arts and skills in as many days and nights. Thereafter, O King, They satisfied Their spiritual master by offering him guru-dakṣiṇā. [SB 10.45.35-36]
Krishna and Balarama learned from Saandeepani is described in Srimad Bhagavatam (10.45):
SB 10.45.32 — Sāndīpani thought very highly of these two self-controlled disciples, whom he had obtained so fortuitously. By serving him as devotedly as one would serve the Supreme Lord Himself, They showed others an irreproachable example of how to worship the spiritual master.
SB 10.45.33 — That best of brāhmaṇas, the spiritual master Sāndīpani, was satisfied with Their submissive behavior, and thus he taught Them the entire Vedas, together with their six corollaries and the Upaniṣads.
SB 10.45.34 — He also taught Them the Dhanur-veda, with its most confidential secrets; the standard books of law; the methods of logical reasoning and philosophical debate; and the sixfold science of politics.
SB 10.45.35-36 — O King, those best of persons, Kṛṣṇa and Balarāma, being Themselves the original promulgators of all varieties of knowledge, could immediately assimilate each and every subject after hearing it explained just once. Thus with fixed concentration They learned the sixty-four arts and skills in as many days and nights. Thereafter, O King, They satisfied Their spiritual master by offering him guru-dakṣiṇā.
From the PURPORT of SB 10.45.35-36:
The Lords learned (1) gītam, singing; (2) vādyam, playing on musical instruments; (3) nṛtyam, dancing; (4) nāṭyam, drama; (5) ālekhyam, painting; (6) viśeṣaka-cchedyam, painting the face and body with colored unguents and cosmetics; (7) taṇḍula-kusuma-bali-vikārāḥ, preparing auspicious designs on the floor with rice and flowers; (8) puṣpāstaraṇam, making a bed of flowers; (9) daśana-vasanāṅga-rāgāḥ, coloring one’s teeth, clothes and limbs; (10) maṇi-bhūmikā-karma, inlaying a floor with jewels; (11) śayyā-racanam, covering a bed; (12) udaka-vādyam, ringing waterpots; (13) udaka-ghātaḥ, splashing with water; (14) citra-yogāḥ, mixing colors; (15) mālya-grathana-vikalpāḥ, preparing wreaths; (16) śekharāpīḍa-yojanam, setting a helmet on the head; (17) nepathya-yogāḥ, putting on apparel in a dressing room; (18) karṇa-patra-bhaṅgāḥ, decorating the earlobe; (19) sugandha-yuktiḥ, applying aromatics; (20) bhūṣaṇa-yojanam, decorating with jewelry; (21) aindrajālam, jugglery; (22) kaucumāra-yogaḥ, the art of disguise; (23) hasta-lāghavam, sleight of hand; (24) citra-śākāpūpa-bhakṣya-vikāra-kriyaḥ, preparing varieties of salad, bread, cake and other delicious food; (25) pānaka-rasa-rāgāsava-yojanam, preparing palatable drinks and tinging draughts with red color; (26) sūcī-vāya-karma, needlework and weaving; (27) sūtra-krīḍā, making puppets dance by manipulating thin threads; (28) vīṇā-ḍamarukavādyāni, playing on a lute and a small x-shaped drum; (29) prahelikā, making and solving riddles; (29a) pratimālā, capping verses, or reciting poems verse for verse as a trial of memory or skill; (30) durvacaka-yogāḥ, uttering statements difficult for others to answer; (31) pustaka-vācanam, reciting books; and (32) nāṭikākhyāyikā-darśanam, enacting short plays and writing anecdotes.
Kṛṣṇa and Balarāma also learned (33) kāvya-samasyā-pūraṇam, solving enigmatic verses; (34) paṭṭikā-vetra-bāṇa-vikalpāḥ, making a bow from a strip of cloth and a stick; (35) tarku-karma, spinning with a spindle; (36) takṣaṇam, carpentry; (37) vāstu-vidyā, architecture; (38) raupya-ratna-parīkṣā, testing silver and jewels; (39) dhātu-vādaḥ, metallurgy; (40) maṇi-rāga-jñānam, tinging jewels with various colors; (41) ākara-jñānam, mineralogy; (42) vṛkṣāyur-veda-yogāḥ, herbal medicine; (43) meṣa-kukkuṭa-lāvaka-yuddha-vidhiḥ, the art of training and engaging rams, cocks and quails in fighting; (44) śuka-śārikā-pralāpanam, knowledge of how to train male and female parrots to speak and to answer the questions of human beings; (45) utsādanam, healing a person with ointments; (46) keśa-mārjana-kauśalam, hairdressing; (47) akṣara-muṣṭikā-kathanam, telling what is written in a book without seeing it, and telling what is hidden in another’s fist; (48) mlecchita-kutarka-vikalpāḥ, fabricating barbarous or foreign sophistry; (49) deśa-bhāṣā-jñānam, knowledge of provincial dialects; (50) puṣpa-śakaṭikā-nirmiti-jñānam, knowledge of how to build toy carts with flowers; (51) yantra-mātṛkā, composing magic squares, arrangements of numbers adding up to the same total in all directions; (52) dhāraṇa-mātṛkā, the use of amulets; (53) saṁvācyam, conversation; (54) mānasī-kāvya-kriyā, composing verses mentally; (55) kriyā-vikalpāḥ, designing a literary work or a medical remedy; (56) chalitaka-yogāḥ, building shrines; (57) abhidhāna-koṣa-cchando-jñānam, lexicography and the knowledge of poetic meters; (58) vastra-gopanam, disguising one kind of cloth to look like another; (59) dyūta-viśeṣam, knowledge of various forms of gambling; (so) ākarṣa-krīḍa, playing dice; (61) bālaka-krīḍanakam, playing with children’s toys; (62) vaināyikī vidyā, enforcing discipline by mystic power; (63) vaijayikī vidyā, gaining victory; and (64) vaitālikī vidyā, awakening one’s master with music at dawn.
You can also find name of these 64 Kalas at footnote at Gita Press Gorakhpur Edition; at Bhagavatam.org.
Other list/name of 64 Kalas can be found in Devi Bhagavatam (12.11):
Quoting English translation of 12.11 from sacred-texts.com:
O King! Now hear the names of the sixty four Kalâs. They are :-- Pingalâksî, Vis’âlâksî, Samriddhi, Vriddhi, S’raddhâ, Svâhâ, Svadhâ, Mâyâ, Sañgñâ, Vasundharâ, Trîlokadhâtrî, Sâvitrî, Gâyatrî, Tridas’es’vsrî, Surûpâ, Bahurûpâ, Skandamâtâ, Achyutapriyâ, Vimalâ, Amalâ, Arunî, Ârunî, Prakriti, Vikriti, S’rîsti, Sthiti, Samrhiti, Sandhyâ, Mâtâ, Satî, Hamsî, Mardikâ, Vajrikâ, Parâ, Devamâtâ, Bhagavatî, Devakî, Kamalâsanâ, Trimukhî, Saptamukhî, Surâsura vimardinî, Lambosthî, Ûrdhakes’î, Bahusîrsâ, Vrikodarî Ratharekhâhvayâ, S’as’irekâ, Gaganavegâ, Pavanavegâ, Bhuvanapâlâ, Madanâturâ, Anangâ, Anangamathanâ, Anangamekhalâ, Anangakusumâ, Visvarûpâ, Surâdikâ, Ksayamkarî, Aksyobhyâ, Satyavâdinî, Bahurûpâ, S’uchivratâ, Udârâ and Vâgis’î. These are the sixty four Kalâs.
Also visit 18 Vidyas.
