1. King Shantanu's dilemma [Episodes 1-2]
King Shantanu the Shagger (whose inability to keep his royal penis in his pants is the proximate cause of all the dramatic events of the entire epic) watches his wife - the goddess Ganga - drown his infant sons, while absurdly melodramatic incidental music plays and the camera repeatedly crash-zooms to his face like a whole season's worth of Colin Baker-era Doctor Who cliffhangers have come at once.
Shantanu is bound by his oath never to interfere in anything that Ganga does, and his inner turmoil is written on his face in the form of some amusingly contorted facial expressions. Ganga's smile after each drowning is the smile of a serial killer, and yet this doesn't ever seem to stop Shantanu from fathering yet further sons with her. Eventually, as she goes down to the river with their eighth son, he finally takes an action to stop her.
2. Bhishma's Terrible Oath [Episodes 3-4]
Saved from drowning, the eighth son of the king is taken by Ganga to be taught by gods, immortals and sages and then returns to his father as the perfect prince and heir to the throne. But Shagger Shantanu's lust gets in the way again when he falls for a fishermanny's daughter, Satyavati. Her father insists that Shantanu may only marry Satyavati if it will be their son who inherits the throne instead of the prince.
Dutiful above and beyond what is expected of a son, the prince agrees to step aside in favour of any other sons his father may have. But the fishermanny is not even satisfied with that and suggests that the prince's descendants may try to claim the throne back from his descendants.
Thus, in order that his father may get a shag, the prince swears a terrible oath that he will be celibate for his entire life, and so is thenceforth called Bhishma: He of the Terrible Oath.
3. The wrath of Amba [Episodes 5 and 70]
Bhishma kidnaps three princesses to marry them to his half-brother (a perfectly acceptable custom within their culture). One of the princesses, Amba, objects on the grounds that she was already in love with, and secretly betrothed to, another manny. Bhishma apologises and sets her free, but her beloved then rejects Amba because his pride has been hurt by his inability to prevent the kidnapping. Rather than be angry at her lover for rejecting her, Amba turns her ire against Bhishma and swears to kill him if he will not make amends by marrying her himself - which he refuses to do becaue of his oath.
Some 65 episodes later we discover, via flashback, what became of Amba after this - she went to Bhishma's own martial teacher, the invincible Parshuram, and persuaded him to fight Bhishma on her behalf. The result was an epic battle between the two, with both deploying devastating celestial weapons, but which ultimately ended in a stalemate. So with Amba still not having achieved her vengeance, she swore to die and be reborn again and yet again until she could somehow be the cause of Bhishma's death.
4. The convoluted circumstances of Karna's birth [Episodes 7-8]
Princess Kunti was taught a magic spell by a sage that could see the future, and he foresaw that this spell would be useful to her later in her life. He explained to her that, when she recited the secret magic words, it would turn her into Superted summon any god of her choosing. But the mischeivous sage did not say what the god would do after it had been summoned. So Kunti tried it out, and summoned the Sun God to appear.
The Sun God then told her he would not leave until he had given her a son, which he then did in the unconventional manner of pewpewpewing Kunti with a special effect until a baby appeared in her arms. Being unmarried, Kunti had to keep the baby's existence a secret, so she set him adrift upon the river to be found and cared for by another family. The baby was Karna, and we have by no means seen the last of him in this series.
5. The even more convoluted circumstances of Krishna's birth [Episodes 10-11]
Evil King Kansa usurped his father's throne and reigned as a tyrant. When the gods themselves prophecised that his sister Devaki's eighth son would cause his death, he imprisoned both her and her husband Vasudev. Not taking any chances that he might lose track of which son is which, Kansa resolved to kill all of Devaki's sons, and so he killed the first six in an unnecessarily gruesome manner by throwing them against the wall of his sister's prison cell.
Obviously his attempt to cheat fate was literally destined to fail, with Devaki's seventh pregnancy being magically transferred to Vasudev's other wife, Rohini, leading to the birth of Balram, while the eighth son is saved by even more direct divine intervention - the gods setting free Vasudev for long enough to deliver his child to safekeeping beyond Kansa's power. This eighth son is Krishna, the avatar of Vishnu.
6. Baby Krishna fights some monsters [Episodes 12-15]
Rather than keep baby Krishna's whereabouts a secret, the villagers of Gokul openly celebrate his coming with a musical number, so that Kansa immediately discovers his location. Advised by an equally evil vizier, Kansa sends a succession of monsters who could have come straight out of an episode of Monkey to kill Krishna. Each one is foiled really easily, because Krishna has the full powers of a god from birth, and so he effectively acts as a living deus ex machina.
At one point he is fighting underwater with Kalia, a giant, five-headed cobra. His friends and family see blood come to the surface, and I was waiting for one of them to say
"Blood! I hope this is not Krishna's blood."
But of course it is Kalia's.
7. Krishna defeats Kansa really easily, while laughing the whole time [Episodes 16-17]
Evil King Kansa convinced himself that if he could defy the gods' prophecy of his death by killing Krishna before Krishna killed him, then he would become immortal. So he set a trap by pretending to repent his evil ways and then inviting his nephews to come and visit him. Krishna and Balram walked into the trap and trivially defeated the attempts by Kansa and his henchmannys to kill them, while all the time laughing in a way calculated to unnerve the king until he tried to run away. But Kansa could no more escape from Krishna than he could overcome him, so that was the end of his reign of terror.
It is unusual to see a dramatic presentation with such a lack of peril for the protagonists, because while on the one paw these episodes seemingly possess all the tropes of a classic underdoggy tail, with our plucky heroes vastly outmatched by the strength and power of the baddy and all his resources as king of the country, yet because of Krishna's godly powers we never feel he is in jeopardy for even one moment.