I was recently reading a Richard Holmes book called Sahib, which talked about this topic. So what follows is a very brief summary of that.

Early on (1750s - 1850s), in southern and central India, warfare wasn't hugely different to how it was in Europe. Many Indian states had armies trained to european standards by (sometimes ex-British military) european experts. The main difference being that the Indian states often had the edge in artillery, as they could produce guns locally. This meant that British tactics had to be much quicker and more aggressive, so as to close to melee distance and rob the cannons of their advantage as fast as possible. I've not heard many accounts of the usual two (or three if earlier) rank line, stationary, being very popular, however anti-cavalry squares were used just the same as in Europe.

As British control began to reach the mountains to the north of the subcontinent, and the enemies faced by were more often tribes than princely states, things started to look more like a form of guerilla warfare that we would recognize today, with the British forces aiming to pacify often already conquered areas of this northern frontier. In these areas the British heavily relied on locally recruited irregular fighters (as opposed to the professional european style sepoys of further south, who made up the bulk of the British and company armies in India), most famously mounted ones, to level the playing field. These men were usually loyal to their charismatic commanders, rather than the British state.

The only example of new equipment that comes to mind was the implementation of regulated water carts manned by (exclusively Indian) men dedicated to that task, this allowed long marches under the Indian sun to be conducted with fewer losses.

Hope that answers your question

This was typed up on my phone, sorry for any mistakes.