The branch that deals with the study of Universe is called Space Science or Space Technology. There are various sub categories like Astronomy, Astrophydics, Cosmology, Stellar Science, Planetary Science, etc under Space Science. Top Colleges for Space Science in India are:
Birla Institute of Technology
Aryabhatta Knowledge Institute of Science and Technology
IIST, Trivandrum
Sri Venkateswara University

There are various levels at which you can join ISRO and at various designations.
After 12th: You can get into IIST (Indian Institute of Space Science and Technology) for their Undergraduate programmes. The admission is based on JEE Advanced scores. Which means you need to have studied Physics, Chemistry and Mathematics in 10+2. You can pick any of their 4 year Bachelor programmes like B.Tech in Physical Sciences, B.Tech in Avionics and B.Tech in Aerospace Engineering, etc. Or you can also pick up their 5 year Integrated courses in Master of Science in Astronomy and AstroPhysics, Master of Technology in Earth System Science, Master of Science in Solid State Physics and Master of Science in Optical Engineering, etc.

Opportunities are amazing for academically brilliant candidates. With a CGPA of 7.5 or higher, subject to the availability of vacancies, you get placed in ISRO. Aero and Avio graduates have better prospects even outside ISRO.

After B.Tech: IIST also offers Postgraduate Programmes like:
M.Tech in Soft Computing & Machine Learning
M.Tech in RF & Microwave Engineering
M.Tech in Geo informatics
M.Tech in Optical Engineering
M.Tech in Machine Learning & Computing
M.Tech in Solid State Technology
M.Tech in VLSI & Microsystems
M.Tech in Materials Science & Technology, etc.

Opportunities depend upon ISRO Placements conducted by centralised recruitment committee.