Indian has many Research Institutes and Universities that one can make scientific visits.  The popular Institute franchises are Indian Institute of Technology (IIT), Indian Institute of Science, Education and Research (IISER). Some of the University/Institute names are:

Raman Research Institute (RRI), Bengaluru

National Centre for Biological Sciences (NCBS), Bengaluru 

Jawaharlal Nehru Centre for Advanced Scientific Research‎ ,Bengaluru

International Centre for Theoretical Sciences(ICTS) , Bengaluru 

Indian Institute of Astrophysics (IIA) ,Bengaluru

Tata Institute of Fundamental Research (TIFR) ,Mumbai

Saha Institute of Nuclear Physics (SINP), Kolkata

Indian Association for the Cultivation Of Science, Kolkata

The TIFR Centre for Interdisciplinary Sciences (TCIS) , Hyderabad

Inter-University Accelerator Centre (IUAC) , Delhi

Physical Research Laboratory (PRL), Ahmedabad

National Institute of Science Education and Research (NISER) ,Bhuvaneshwar

Raja Ramanna Centre for Advanced Technology, Indore

Indian Institute of Space Science and Technology ,Thiruvananthapuram

Jawaharlal Nehru University (JNU) , Delhi

University of Delhi 

University of Hyderabad

Banaras Hindu University

Institute of Mathematical Sciences (IMSc) , Chennai

Chennai Mathematical Institute (CMI)

S. N. Bose National Centre for Basic Sciences , Kolkata

Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune

Institute for Plasma Research (IPR) , Gandhinagar

Harish-Chandra Research Institute (HRI) , Allahabad

Institute of Physics (IOP), Bhubaneswar

Homi Bhabha National Institute (HBNI) Mumbai

Homi Bhabha Centre for Science Education (HBCSE), Mumbai