Amity University, Gurugram

These are the major entrance exams in India for Msc or integrated Ph.D. (or direct Ph.D. in some cases) in India.
JAM
Opportunities abroad (Physics GRE, General GRE, IELTS and TOFEL etc)
TIFR GS(Integrated Ph.D. paper)
University entrance tests.
MS by research programs
JEST
TIFR Hyderabad
GATE
CSIR NET
The three exams JEST, GATE, and NET have syllabus of Msc+Bsc. All other mentioned above are conducted on Bsc level syllabus and are easier for us Engineers with less physics background to crack.
JAM exam: This is conducted by IITs as an entrance test to Msc and integrated Ph.D. in IITs and IISC. Also, some IISERs take jam rankers to integrated Ph.D. This is the best of all options for slow learners. For those who can't spend a year preparing for entrance tests, this will be a boon.Even if you prepare perfectly only up to 12th class level physics, you can get a rank under 500 easily. So, even if you are preparing for JEST, I strongly recommend keeping this as a backup option at least.
Caution: IITs and IISERs ask for 2 years of physics related subjects in bachelors as compulsory for JAM rankers. I heard that mechanical guys may not meet these criteria.To confirm if you will meet the eligibility criteria, mail those institutes and inquire. I think IIT Kanpur and Bombay give an exception.
TIFR GSexam: TIFR is one of the elite research institutes in India if not the best. Even better than IISC as far as physics is concerned. If you opt for integrated Ph.D. paper, Your questions will be from Bsc syllabus but the no.of questions will be less and look for your deep understanding of a concept. Very few people get selected this way. The interviews will also be very thought-provoking. You need to be very strong with your fundamentals. All TIFR institutes will take students through this.
Opportunities Abroad (Physics GRE, General GRE, IELTS and TOFEL etc):I personally feel if you can fund yourself, then this is the best option.Because switching career is very easy thing abroad and the opportunities you have there are numerous and the research happening even in the mediocre colleges there is comparable to good colleges here. General GRE with a combination of TOFEL or IELTS is necessary for the USA and Physics GRE is recommended. As for the European countries, only IELTS is enough (Sending GRE scores might add to your profile but is not mandatory). The tuition fee is zero in some countries like Italy and Germany.