Thank you for your interest and help Pedrov.
I have done a lot of reading around the whole period and like yourself have found lots of detail on who was at war with whom and why but only a few direct references to what type of units made up their militaries. This is what I know in general terms, please let me know if any of it is misguided or if you know more:
Generally
Firearms: There was a lot in the way of firearms being used at this time. I've found a few references to 'rockets', lots about musketmen and seen Mughal paintings of people playing with fireworks. But nothing specific to move forward on for the scenario in creation so I'm not sure what form these firearms took on the battlefield.
Cavalry: Lots of references everywhere to cavalry. Many kings used 'superior cavalry' tactics and 'superior heavy cavalry'. But what was the difference between these and the inferior cavalry? What did they look like?
Ships: Indian ships were big and slow with very large capacities. They followed either Chinese ship building designs or Arab / Ethiopian designs.
Specifically:
Rajputs: They offered the Mughals stiff resistance but with what kind of military? I have been to Rajasthan many times and they have camels everywhere. I've ridden them too

. I presume these wonderful beasts were used extensively but I don't know for a fact.
Mughals: Superior Cavalry and plenty of firearms and artillery. Again don't really know names or the right look.
Golkonda: With the Golkonda mines producing the biggest diamonds in the world these guys were very wealthy and with good connections to the Safavid Persian Shah Tahmasp. I have no idea what their military was made up of though and whether these relations contributed to military make up..
Bijapur & Ahmadnagar: As with Golkonda.
Safavid Persians: Shah Tahmasp was evidently a powerful shah. He was kind enough to 'lend' Humayun a small fraction of his army to go and retake Delhi. Not a mean feat. But I have no idea what the army consisted of.
Vijayanagara: Yes these guys were tough barstewards. I expect they used elephants extensively but can you confirm this please? Were these Mounted Mahuts with bow in hand? I can't imagine a sword would do much good 10-5ft above the battlefield! What kind of infantry? Did they use firearms widely like the others?
European traders: Plenty of cannons, walls and musketmen. I know, I'm from Goa originally and I've been to the forts. But was this all they used?
Bengal: I am aware that Shah Sher Khan was a military whizz-kid and defeated many of his rivals. I suspect he used similar methods and units as the Mughals - again don't know this for fact.
Ceylon, Burma, Nepal: So far I know nothing about these guys during the 16th and 17th centuries. Gurkahs for Nepal I guess, they have a long history which may well start around this time.