Describe and analyze the continuities and changes of Hinduism (India) from the beginning of the religion or belief system up to 600 CE?
Records of the earliest Hindu beliefs can be found in the Vedas and the later Brahamanas, Aranyakas, and Upanishads (philosophical commentaries on the Vedas), which record beliefs that date from between 1500–500 BCE (although these was not written down until much later; they were part of an oral tradition). There is also no definite starting point for Hinduism and we know very little about the religious traditions of the Indus Valley Civilization, which existed before 2000 BCE. The Indus Valley Civilization did place emphasis on ritual bathing, sacrifice, and goddess worship; these connect it to Vedic and later Hinduism. However, the Indus Valley religion shares almost as much with other contemporaneous ancient religions.
While the language of the Indus Valley Civilization is still unknown (the script is made up of symbols that have been partially deciphered), the Hindus of the Vedic period (1500-500 BCE) spoke Sanskrit. We consequently know more about the religion of this period. Hinduism in the Vedic period centered around ritual sacrifice (not just of animals but also foods like milk) and sharing of ritual meals. There was a developed (and hierarchical) pantheon of gods and the rituals were performed in their honor. Some of these rituals, particularly those to do with fire (agni), continue until the present day. The Vedas are comprised of the Rig Veda (oldest of the vedas), which is a collection of hymns to the Vedic deities, the Sama Veda and Yajur Veda, both of which contain melodies for chanting the hymns and sacrificial formulas, and finally the Atharva Veda which is compilation (including hymns from the other Vedas) and which also contains magical charms
The major Vedic gods were associated with elements of realms of the universe: Soma (plants, earth), Agni (fire), Brhaspati, Indra, Vayu (wind), Maruts (storm, planets). Dyaus (sky), Varuna (guardian of the cosmic law, rta), his counterpart Mitra (night), and the pervader Vishnu.
Buddhism came into being circa 500 BCE (Buddha died circa 400 BCE) and this period marked a major shift in Hinduism. The move was from the Vedic Age into the Epic and Puranic Age which lasted from 500 BCE until around 500 CE. The Mahabharata (of which the Bhagavad Gita is a part) and the Ramayana (two of the most important 'Hindu' texts today) were composed during this period. The later Upanishads were also composed at the beginning of this period and they emphasized devotion to a personal god and the practice of yoga rather than renunciation.
There was a shift away from fire sacrifice and such rituals and a move towards idol worship in temples. The Vedic rta — the cosmic law as well as the law that regulated mankind — was replaced by the notion of dharma (dhamma in Pali). There were a series of texts that explained dharma and this notion became central to Hinduism and continues to be so until the present day. There was also a change in the pantheon of gods with certain gods (like Agni, who was extremely important in the Vedic period) losing importance and other gods being introduced. Hinduism, at this time, was also divided into followers of Vishnu and followers of Shiva, a division that also persists until today The early puranas, law codes, and philosophical systems such as Vedanta also stem from this period.
(See also the useful timeline provided in the reference links)