JEE Main

If your general rank is 6213, then u'll easily get IT at IIIT Allahabad because of reservation quota.

Nowadays coaching is required for the preparation of all exams. But if you do not want to join a coaching then you should focus on what is taught in school and attend the classes regularly.
The school will prepare you for the boards. for JEE Mains you should thoroughly prepare your school syllabus of class 11&12 as the syllabus for JEE is same. You should solve previous year question papers of JEE and can also buy some books for JEE preparation. Your basics of 11th and 12th class should be clear.
All the best.

Sorry to say but you are not eligible for any of the government colleges including NITs, GFTIs based on your marks.

Sorry to say but maximum of three times a candidate can appear in JEE Mains but for JEE Advanced, this is restricted to just 2 chances. In short you can't appear in JEE Advanced in 2017.

Yes, you are eligible. You need to visit the campus.

Approximately 250 to 290 marks.

Hi,
With this rank, the chances of getting CSE in JECRC will be comparatively less. Try going for other branches.

Can you tell me your rank?

You have to take JEE Mains exam. If you qualify JEE Mains with good marks then can easily get admission in an NIT.
Yes, student with less then 70 percent marks can take admission in NIT through JEE Mains but you have to score more if you want to get admission in a good NIT.