M.Sc

You could try Indian Institute of Astrophysics, Bengaluru. It is one of the oldest institutions in India (1786 AD) and right now, the premier institute for Astrophysics (in my opinion). They offer three courses:
M.Tech, Ph.D. Dual degree GATE written test)
M.Sc, Ph.D. Dual degree JEST written test)
PhD. Currently they operate out of their campus in Koramangala, Bengaluru, though the CREST campus - Hosakote, Kodaikanal Solar Observatory, Vainu Bappu Observatory, the Gauribidanur radio telescope and the Handle Observatory are under them. For more information visit Indian Institute of Astrophysics
2. You could also try IUCAA (Inter University Center for Astronomy and Astrophysics), Pune. Visit IUCAA (qualifying exam: JEST)
3. If you are interested in theoretical Sciences, You could try TIFR (Tata Institute of Fundamental Research) or JNCASR (Jawaharlal Nehru Centre for Advanced Scientific Research) or RRI (Raman Research Institute) (qualifying exam: JEST or GATE)
4. IISc (Indian Institute of Science, Bangalore) provides courses related to astronomy. The advantage is the beautiful campus of the best Institute of India. GATE or JEST or written test)
5. Not really Astronomy, but Indian Institute of Space Science and Technology (Home : IIST) runs courses in Engineering related to Aerospace and Avionics. In-so-far as job opportunities are concerned, Research and teaching would be primary. For example, the Handle telescope in Leh is controlled through CREST, Hosakote. So that's a more technical job. IIST, however is quite different from the rest. It is a company (ISRO) backed institute and the placement might be different. The institute is possibly the most beautiful one you will see.