Computer Engineer

You have following options to become IT engineer:
(a) B.Tech CSE / IT
(b) BCA-MCA
(c) B.Sc-M.Sc CS/ IT
(d) Diploma in Computer Science
(e) DOEACC Certifications
(f) Graduation degree with computer courses etc. You can choose any of the path as per your preferences and interests.

If you want to further study with your job, then you can choose distance mode course for further course. There are many institutes which provide distance learning in your field.

For fresher, it will be around 3 to 4 lakhs. If you have an attractive resume, it may be bit high then.

You have to do B.Tech in computer engineering or information technology from any good reputed college. I will suggest you to join any college in Bangalore that will be the best.

It's great to hear you didn't mention you want to do engineering instead. You can go for courses of NIIT, DUCAT, etc. They will introduce you to the world of IT better than any engineering college.

B.Sc (CS) is better than BCA.

Following are my recommendations in order of precedence: B.Tech CS/ IT, BCA-MCA, B.Sc-M.Sc CS/ IT, Diploma in Computer Science and Engineering.

If you want to become a software engineer then following are the options in front of you in order of precedence:
(a) B.Tech CSE or B.Tech IT
(b) BCA-MCA
(c) B.Sc-M.Sc CS or B.Sc-M.Sc IT.
Nowadays it is difficult to get a software engineer job if you are engineer with a degree such as Mechanical, Civil, Chemical etc. not from above mentioned IT background degrees. Mass Media qualification is for becoming journalist.

Web designing is a saturated market with lot of competition and also it's a narrow field whereas Software Engineering is a wider field with wider scope and opportunities. Web Designer may not able to do Software Development but software engineer can easily work as Web Developer & Designer. You are advised to choose Software Engineering over Web designing for long term career growth. You may opt for B.Tech CSE/IT, BCA-MCA or B.Sc-M.Sc CS/IT.