In terms of information, I doubt I can improve on Shion's answer. However, I'll add some comments.
First, a disclaimer. I don't know much about Indian academia, but I was a grad student in math for a while, a long time ago, at TIFR. It did not go well. I also know (knew, perhaps) a fair number of Indians who doing research mathematics.
First India nominally has a lot of universities. However, the vast majority of them are not research universities. At least in mathematics, my impression is that most of them don't count for much on an international research scale. That, of course, does not necessarily mean you will get a bad mathematical education there. It all depends on personal experience, including ones advisor (sometimes out of the way places have good research people), the local atmosphere, plus the internet, with sites like mathoverflow, is definitely having a levelling effect.
In any case, the important places for mathematics, as Shion says, are the handful of (mostly government sponsored) research institutes. There are also the various IITs, which do have math departments, which for the most part are better than the "regular" universities, and probably not as good as the research institutes.
If you go to study at the research institutes, I think you can get quite a good mathematics education there. I don't think it is any worse than the education you would get at a good Western university, and in some ways it may be better. For one, thing you will experience less distractions. See below. For example, the School of Maths at TIFR was, and I expect still is, quite a high powered research place, and has good connections with the international research community. There are regular visitors from abroad to give talks and visit. TIFR Faculty travel abroad freqently on visits, giving talks, taking sabbaticals and so on.
However, some caveats.
First, as Shion says, they are mostly smaller places (though I think the TIFR school of maths is quite substantial). Second, as a math student, you are likely to find yourself extremely socially isolated. These places pay very badly, regardless how good a mathematical education they impart. They also tend to be located in expensive metropolitan areas. For example, the TIFR is located at the tip of South Bombay. Bombay is possibly the most expensive place in India. You won't have enough money to do much of anything. Plus, of course, you'll be very busy studying.
These institutes are additionally typically not part of a university campus, so you aren't surrounded by the considerable variety of people and happenings you experience in a regular university environment. You'll be stuck with your fellow students, postdocs and faculty, for the most part.
These issues are to some extent true of a mathematics student anywhere, but they are, in my opinion, more extreme in India. I was a grad student in the US, and it was not nearly so isolated.
There are some advantages for an Indian citizen to being in an Indian research institute as opposed to say a European or US university, however. First, you won't have to put up with being treated like a third-class citizen, and being constantly pushed around by government and university bureaucracy. Second, you'll have all the time in the world to do your research. Indian research institutes, unlike Western universities, don't bother their students with things like teaching duties. Since they aren't regular universities for the most part, they don't have much by way of students anyway. There will probably be opportunities to teach courses if you want, though, if only at neighboring colleges.
So, to summarize if you don't mind going years at a stretch doing not much but study maths, you'll be fine.
A final word about areas represented. I think the math research institutes skew heavily towards theory. I think there is some CS, but again theoretical. The TIFR for example is entirely theoretical, though I believe there are some people there who do consulting. My impression is that the IITs are more applied. If you are interested in things like machine learning, those might be better places to look at. Areas like statistics are also poorly represented. There is ISI, but not much of anything else at the level of statistics graduate study.
Some of this may be factually inaccurate. I am sure some of it is biased. Factual corrections appreciated; I am always happy to learn.