Hello Isshaan,

Some of the top research institutes in India are,

Tata Institute of Fundamental Research (TIFR), Mumbai

Indian Institute of Science (IISc), Bangalore

All India Institute of Medical Sciences 

Post Graduate Institute of Medical Education and Research, Chandigarh

Indian Veterinary Research Institute, Bareilly

Centre for Studies in Social Sciences, Kolkata

Tata Institute of Social Sciences (TISS), Mumbai

Indian Agricultural Research Institute (IARI), New Delhi

Indian Institute of Mass Communication (IIMC), New Delhi

Hope this helps.

All the Best.