India

Trends on various engineering branches scope and opportunities varies with multiple parameters such as Economic growth, Geo Politics, innovation & technology changes, shortage of skills, quality of education, changes social capital etc. Therefore, predicting trends is very difficult. Student must choose his branch based on his/her interest to excel in professional life.
This decade belongs to Electronics and Telecommunications. Electronics and Telecommunications will continue to impact all walks of life including business. Next wave of Industrial revolution is automation and Electronics will play a vital role in it. ECE engineers can be employed in all type of industries including IT. Computer Science is the facilitating technology for all business and personal activities. Indian IT industry is worlds largest outsourcing hub. Computer Science engineering is dominating the job market for last 15 years. Indian Computer industry has started showing early sign of decline due to outcry against outsourcing in western world. Therefore, Electronics &Telecommunication is best option with respect to career opportunities and scope. Computer Science will continue to good career option for some more times in the field of Engineering. Mechanical is an ever green branch for last 50 years and will continue to remain same in next 50 years. Civil Engineering is an age old profession and with Government of India's focus on infrastructure scope and opportunities for Civil Engineer will be very high for next few years. Most of the jobs are in field. India has very well developed Chemical Industry including Petroleum & gas both in Government and private sector. It is core sector with lot of Government and big corporate houses support. Chemical Engineers have good scope in India and abroad. Its market is little seasonal and has heavy dependence of Geo political environment. Electrical and Electronics Engineering (EEE) is a core field of engineering but majority of EEE graduates are treated as Electrical engineers.
My recommendations in order of precedence are ECE, CSE , Mechanical, EEE, Chemical and then Civil engineering if student does not have any specific area of interest.

Tamil Nadu has many good coaching centers like:
Aakash: The teachers here are good, but it's very expensive if you compare other JEE institutes some of which are better. It's good for medical preparations though.
FIITJEE: They take a test based on which they divide you into batches, the students performing better are given better batches which sometimes demoralizes the students who dint get into good batches. They only concentrate all attention and focus on these students which I personally disapprove of.
Brilliant Tutorials / Career Launcher : Their performance has been depreciating for a while , they provide you basic mentorship and have good test series , though there have been many cases where people have paid the entire coaching amount but not received any coaching or refund.
BYJU'S CLASSES: They have been doing really well in last couple of years as 161 out of 212 students made it through IIT's last year. I have heard they make classrooms a fun interactive session for their students which works in their favor. Also they have been known to return the entire fee amount if the student does not find them good enough for JEE. The only downside is that they accept only a certain number of students and the batches fill up really fast.
Source - https://www.quora.com/What-are-the-best-IIT-JEE-coaching-centers-in-Chennai

This is a list of coaching centers available:
But this list is not ranked, it just gives all options available. So research a bit before enrolling. Check for reviews on net and talk to some students in those institute.
1. Bansal Institute for Excellence
Bansal have a best coaching classes in Ahmedabad and with the good teachers and faculties.
2. Brainquest Science Academy
Brainquest academy focused on the basic concepts. The faculty totally focused on concepts learning approach Brainquest academy discipline and on time. Due to quality management now Brainquest offers coaching in many courses but its focus has been on IIT JEE since inception.
3. Career Point
Career point is good is good institution in Ahmedabad he teach students by good books and updated syllabus. All teacher are good and experienced in field. Career point has been guiding students for IIT JEE since edges and performance of students at coaching is good.
4. Precept
Precept Academy is effort you put in will determine your successful life in your career and personal life. The teachers have good subject knowledge and are guiding you well.
5. Abhyas
Abhyas is perfect in IIT-JEE, JEE Mains, AIMT by the best faculties, they have experience ever had to introduce to your hidden talents. Although Triumph Academy focus was on CAT coaching's but its gaining popularity brought it in IIT JEE coaching and its doing good for students who are preparing for IIT JEE.
6. Abhyas Practice for IIT-JEE
7. .S.N. Enterprise
8. B.R. Tution Zone
9. Aakash Educational Services Ltd.
10. Nirva Maths Group Tuition
Again: Don't use this as final list, do your own research before enrolling.