Was armour even used by common soldiers? Which type of bows were used? Was horse archery ever a thing? Until when were the chariots used? If elephants were not effective then why were they used so commonly?
I have so many questions, answer to any one of them is appreciable.
What kind of the arms and armour were used in Ancient India (before 800AD)?
For the sake of convenience I'll try and limit the time span we are trying to analyse to post 500 BCE and pre 800 AD. Now the sources for this time period are texts such as the Vedas or the Epics and later texts such as Puranas, as well as archaeological evidence.
If we look at the weapons and armour used by the Indo-Aryans, upon their arrival, we find references to bows, quivers, arrows, chariots and spears as well as chain mail. We can safely say that chariots, horses, archers and light cavalry were the dominant arms of armies of the Indo-Aryan states during the period of the Mahajanapadas.
Here is a translation of a Hymn from Book 6 of the Rig Veda, it's Hymn 75
Keep in mind however that during the era of the Mahajanapadas, chain mail was not affordable for all, in fact such armour would perhaps only be conceivable for nobles of the tribal and later feudal structure that exited in the ancient and classical age. The most common of all soldiers would probably wear one or two layers of clothing, if affordable, or wouldn't cover their torso. They would either be foot archers or infantry armed with spear and axes.
In later centuries, with the rise of centralised Empires such as the Mauryans, we find extensive use of elephants, and archers mounted on elephants, while still it's not certain whether or not the common soldiers wore layers of clothing to cover their torsos, what's certain is that the nobility and military officers etc. were able to afford good armour such as chain mail. Hence, elephants during this period were considered an integral part of the armed forces.
During the Gupta-Vakataka age, roughly 300 CE to 600 CE, we find that given the nature of the enemies that the Guptas faced such as the Hunas, the Kidarites, the Sakas etc. that the Guptas shifted the focus from slow moving armies of the past which included hundreds of elephants if not thousands to faster, more responsive and better equipped heavy cavalry armies. Keep in mind "heavy" here is not to describe the size of the horse or the rider, but rather the arms and armour of the horsemen.
Such coins as well as sculptures such as these suggest the existence of better armoured, horse cavalry which made use of scale armour and was armed with composite bows and swords. Also, swords in this period (800 BCE-800 AD) weren't curved like later tulwars but rather were straight and double edged like this Khanda
These trends of large numbers of cavalry in Indian armies would continue. And while elephants still featured in the armies of states that succeeded the Guptas, especially in the regions of Bengal, Orissa, the importance and applications of cavalry were understood, especially in the region later knows as the North Western Frontier. However, in later centuries, especially after the fall of the Gupta Empire, horse archers weren't a major feature in North Indian armies yet, cavalry remained. Overall the weapons of Indian armies, especially common soldiers varied from axes, maces and usually spears and shields. Swords being costlier would usually be either provided to units of cavalry supported by imperial treasuries by the ruler themselves or local feudal lords would make sure to arm their cavalry with swords, as per their administrative and military obligations. Cavalry, usually carried lances or spears and swords. While there may have been horse archery during the period between 1500BCE - 800 BCE, this largely disappeared until the Gupta Era, due to a change in societal structure and organisation. The nomadic-pastoralists, that were the Indo-Aryans, shifted to being sedentary city and village dwellers. Therefore, infantry came to dominate armies and the traditions of horse archery began to disappear.
While these trends of horse archers in armies were revived during the Gupta Era. It was owing to very specific circumstances and opposition that the Guptas faced and with their decline, these trends once again disaapeared however the use of cavalry continued to dominate the military trends of North Indian states West of the modern day Bihar region, between Kabul and Kashi.
Which type of bows were used?
As per the description of Arrian, a Greek historian and military commander, as well as the author of The Anabasis of Alexander Indian archers used longbows, probably made of either wood or bamboo, which were as long as their users were tall. These bows would be rested in the ground and would be held in place firmly by the archers who held them between their toes. Next the arrow was drawn on the bowstring and shot and according to Arrian, they were lethal enough to pierce a shield.
Until when were the chariots used?
By the time of the Tripartite Struggle it seems to have disappeared altogether.
If elephants were not effective then why were they used so commonly?
I'm assuming you mean pre - 800 AD here as well. In this case, the reasons were manifold.
The first being intimidation. Fresh infantry and especially poorer levies would be unwilling to charge against an army which boasted large numbers of elephants.
The second being a vantage point. Even on a flat surface an elephant's back provided a good vantage point to observe the battlefield and to command various forces. It was also a good place to mount 3 or 4 archers, who could constantly pepper the enemy with arrows from a height.
The third being their shock value. When a hundred elephants charge in one direction at once, it creates a psychological effect upon the enemy forces which may convince many fresh recruits to rout and retreat. Plus, unless an army is prepared or well led, there really aren't many ways to combat 100 or more elephants charging towards you.
I've provided several sources above, but for further reading I suggest these :
"Warfare in Pre-British India, 1500 BCE to 1740 CE", by Kaushik Roy
"Alexander the Great : selected texts from Arrian, Curtius and Plutarch" by Tania Gergel
"The Classical Accounts of India" by R. C. Majumdar
"The History and Culture of the Indian People" by RC Majumdar, Volumes 1, 2 and 3
