The three stories from three yuga connect the victory of good over evil.
Vijay Dashami
Vijay means ‘Victory’ and Dashami is the ‘Tenth Day’. This is the day when Goddess Durga defeated the asura king Mahishasura. Mahisha is a buffalo and asura is a demon or people from the underworld.
In the puranic times, Rambha the asura king fell in love with a water buffalo (a cursed princess herself) and Mahishasura was born. He has immense occult power like shifting forms. He did hard penance to become immortal and pleased Brahma. As Brahma said he cannot give the boon of immortality, the asura sought a boon that no man or animal can ever kill him.
Mahishasura soon was undefeatable, and his tyranny spread across the three worlds. He charged Amravati, the capital of devas, and drove the gods out. The devas consulted Brahma, who could not change the boon. Lord Shiva, Lord Vishnu, and all the gods went to war with their mightiest weapon, Mahishasura took different forms and rebounded.
The trinity of Lord Brahma, Shiva, and Vishnu consulted to find a way out. They agreed to combine all their powers and that of all devas to create an invincible woman, to destroy the evil asura. You can read the story here. She was Lord Durga.
Durga roared to challenge Mahishasura to a war. Initially, he ignored her and sent his army. Once the army was defeated, he took notice and tried to lure her to be his companion and queen. When she rejected this proposal, the asura king became ferocious and attacked Durga. After nine days of intense fighting, Mahishasura was slayed. Devas got back their capital and world relieved of the tyrant on the tenth day. Durga is called Mahishasura Mardini, the destroyer of Mahishasura.
This day is celebrated as Vijay Dashami. The Devi is worshiped during the Navratri and symbolically returned back by submerging her idol in water bodies with lots of fanfare.
Dussehra (Ramayana)
Dus means ‘ten’ and hara means 'defeat', thus Dussehra signifying the victory of Lord Rama over the ten-headed asura king Ravana.
Lord Rama was the eldest prince and heir to the throne of Ayodhya. Due to a boon his father Dasaratha gave to one of his beloved wives, Kaykaye, Rama had to go to exile for fourteen years. His wife Sita and brother Lakshmana went along.
While in the forest there was an incident where Lakshman disfigured the asura princess Surpanakha, sister of asura king Ravana. Ravana abducted Sita to seek revenge. Rama & Lakshmana along with the mighty army of monkeys built a bridge across the ocean and challenged the asuras. But Ravana was no ordinary king, he had 10 heads, was one of the most learned brahmins; the storehouse of all knowledge, master of occult & magic, had a large army, and had defeated Gods. When Ravana did not acknowledge to return Sita without a fight, the war was inevitable.
Just before the battle, Rama set in a yagya to get the blessings of Lord Durga. This puja needed 108 blue lotuses, but there were only 107 available. To complete the required count, Rama took out an arrowhead to cut out one of his eyelids, as they resembled a lotus. The Maa Goddess was pleased, appeared, and blessed Rama.
The war continued for many days, with no results. Although many of the warlords of Lanka were killed one after another, it took very long and damaged the morale on both sides. Ravana came to the war-field only after all his sons, brothers, and generals were slayed. He was not easily dominated, due to his warring skills, weapons, and occult powers. After having received huge damages to his army did Rama use Brahmastra to hit Ravana in his heart to defeat him.
The day Rama killed Ravana, we celebrate that day as Dusshera. Large idols of Ravana are symbolically burned with lots of fanfare at the end of Ram Leela celebrations.
Ayudhya puja (Mahabharata)
Ayudhya puja is the worship of instruments that one uses integral to their lives. It is worshiping a weapon for the warrior, plow by the farmer, worshiping Saraswati for the student, worshiping the vehicles by the driver, and the likes.
The legend goes like this. When the Pandavas were defeated by the Kauravas in the game of dice, with the trickery of Shakuni, they had to go for vanvaas (exile from their kingdom) for 13 years. A condition was put that for 12 years they will be in the forest and the last year needs to be agyaat (undercover, where no one can know their identity), and if they are identified the cycle of vanvaas will restart.
The Pandavas prepared for defeating the Kauravas in the 12 years in the forest. In the final year, they hid in Viraat kingdom in various menial roles in disguise. The brothers hid all their weapons on a Shami tree. After their 13th year in exile was over, the Pandavas on Vijayadashami day, retrieved their weapons from the Shami tree and worshipped them. They went to Hastinapur to claim their stake. Eventually, they went on a war against the Kauravas on the battlefield of Kurukshetra. Pandavas were victorious. Pandavas returned victorious to Hastinapur on Vijayadashami day.
Since then it is believed that this day is auspicious to begin any new venture.