There are different rituals followed in different parts of India.
Durga Puja:
Goddess Durga killed the demon Mahishasur during this time, representing the victory of good over evil. During all nine days of Navratri and Dussehra, people worship Goddess Durga.
Durga Visarjan:
In the eastern parts of the country, particularly in West Bengal, Durga Puja celebrations precede Dussehra. On the tenth day of the Pooja, that is, the day of Vijaya Dashami, idol of Goddess Durga is immersed into the nearby river or lake, by the devotees. The devotees bid farewell to the deity very ceremoniously and this custom is called Visarjan . In these region, Vijaya Dashami is celebrated to commemorate the victory of Goddess Durga over Mahishasura the demon.
Vidyarambam
In the Southern states of Tamil Nadu, Karnataka and Kerala, the tenth day or Vijaya Dashami is also the day of Vidyarambam (beginning of study). People generally worship Goddess Saraswati,(Goddess of Wisdom), on Vijaya Dashami. The day normally starts with worship of books and learning materials. It is considered very auspicious to start any form of learning on this day.
Sindur Khela:
In the state of West Bengal, married women play among themselves with vermilion (sindur) as a part of Dussehra ritual.
Weapon worship:
There is also a ritual of worshiping weapons, pens and other instruments that are a symbol of fighting injustice.
Wagtail search:
In some parts of India there is a ritual of finding a wagtail on Dussehra day. If one is able to find a wagtail among flowers, elephants, cows, horses, snakes- it symbolizes good-luck. While if it is found in ashes, bones, then it is believed that an evil will follow, for which medicinal bath is taken.
Worship- instruments of Profession:
Across North India, this time is considered very auspicious to polish one’s professional instruments. Drivers, clean and polish their trucks and buses. They put Roli and draw swastika on them. This ritual is also performed in army and other forces.
Banni leaves collection:
In parts of Uttar Pradesh ,this festival is associated with the ancient legend of the young Brahmin boy Kautsa s act of distributing gold coins among the poor people .Thus, people of Uttar Pradesh, collect leaves of Banni trees, treat them as golds and use them to greet their friends and relatives. During this time people decorate the entrance of their houses with torans and flower studded strings. The leaves of the Apta tree are collected and exchanged among friends and relatives as gold.