IIT

I'll prefer you to join any other coaching institute like Allen and Resonance as the coaching institute mentioned by you is also quite good. But the main problem is that their passing ratio is not that good.

I'll suggest you to opt for FIITJEE as this coaching institute has good faculty and also has the easiest way to teach students the most difficult questions.

IIT-Bombay
IIT-Delhi
IIT-Madras
IIT-Kanpur
IIT-Kharaghpur.

For NIT and IIT, you can get more deep knowledge than state board universities. You can get good placement and learn more new things.

The best courses you can pursue from IIT and NIT are computer Science, mechanical engineering, electrical engineering and some other.

If you have interest in pure science and want to become a scientist or researcher, then go to IISc. If not you want to do a job then go for IIT. It all depends on your interest.

Are you asking about JEE Mains preparations? Then, yes, you can study through books of NCERT, maths. You can go for RD Sharma and for physics, you should go for SL Arora. For Chemistry, I would prefer Dinesh Publications. In the link below are the suggestions of some toppers of JEE Mains:
https://www.shiksha.com/b-tech/articles/crack-jee-main-2016-with-exam-prep-tips-from-last-3-year-toppers-blogId-11714
Below link gives you some preparation tips for JEE:
https://www.shiksha.com/b-tech/articles/10-tips-towards-your-jee-mains-advanced-preparation-blogId-11721.

It totally depends on following:
1. Cut off of colleges.
That mostly depends on number of students giving exam.
2. Seat available in colleges
3. If you have any quota, e.g.-handicap, sportsmen etc.

NPTEL is the best website for free IIT lectures. It provides good quality of content/notes from IIT.

For entering into the IITs, you need to qualify two the toughest examination I. e. JEE Mains and JEE Advanced. And for cracking these two exams, you need some best study strategies:
1. Understand each topic clearly.
2. Do lots of hardwork and practice more and more questions.
3. Make smart and good notes.
4. Revise all the formulaes.
5. Solve as many as previous year question papers.