Connect with right people and academies
There are programs to spend a semester or a year at an American university as part of your studies at a university in your own country. Those would be administered through your home institution, and you would have to contact them for more information. You can also come to an American college or university intending to get a degree from that institution. You would have to apply directly to each institution in which you were interested. Almost every institution would accept applications from foreign students.
There are thousands of colleges and universities in the US. All the states and most large cities have a university, usually more than one university campus, and there are private institutions large and small. Here is one list, I cannot recommend any particular one:
http://www.mit.edu/people/cdemello/univ-full.html
There are several good universities in India for astrophysics. Try looking at these sites for some more information:
http://en.wikipedia.org/wiki/Inter-University_Centre_for_Astronomy_and_Astrophysics
http://www.tifr.res.in/scripts/homepage.php
http://www.cs.wisc.edu/~sodani/indian-colleges.html
http://ccsd.cnrs.fr/PhysNet/india.html
I found these by entering "India University Astrophysics" into a search engin.
India has an active space program. Take a look at the Indian Space Research Organisation web pages:
http://www.isro.org/
Jay and Amy
for Ask an Astrophysicist
The following aditions to the above answer was provided by Sudip, an Indian astrophysicist who voluntereered his time for the "Ask an Astrophysicist" project while he was working in the US.
In India, the salary of a scientist/professor is in the middle class level. The salary of a computer programmer/engineer is several times higher than that of scientists/professors. The difference is much more than it is in the US.
In India, there are about 20000 colleges, about 250 universities, many private institutions (mostly for engineering and management), teaching/research institutes (such as IITs, IISERs, etc.), many laboratories, and so called "elite" research institutes like TIFR, IISc, etc. People with academic (research) interests can go to the US either for doing Ph.D. or for doing a postdoc. The latter, unlike the former, does not require an exam (GRE). Those who do Ph.D. in India normally gets a good scholarship without doing a teaching/research assistantship.
Astronomy is generally not taught in universities (except some like Delhi, Osmania, etc.). Those who want do Ph.D. in astronomy in India should first do a bachelor degree in physics (or engineering with inclination to physics) or a Master degree in physics, and then join (by clearing the entrance test) one of the research institutes that offer astronomy, such as Tata Institute of Fundamental Research (TIFR), Inter-University Center for Astronomy and Astrophysics (IUCAA), Raman Research Institute (RRI), Indian Institute of Astrophysics (IIA), Indian Institute of Science (IISc), etc. Students normally do very good Ph.D. works from these institutes. However, one probably should go to the US for postdoctoral works.
Getting a permanent scientist/professor job (especially in a good research institute) is very tough in India, as it is in the rest of the world.
There are some observational facilities in India: GMRT (radio), Astrosat (X-ray; upcoming), Tauvex (UV; upcoming), some optical telescopes, etc. Also Indian institutes have collaborations with some international facilities.