These are all correct sentences. I have corrected where appropriate:
Did anybody tell you what happened? HAS anybody told you what happened?
I have BEGUN to study English.
I learn English on the distance course. I learn English via the distance course.
Usually I learn in the evening.
I revise the previous material and then I go ON TO new material.
I revise the previous material and then I go on to new stuff. That is how I would say it, just to avoid repeating 'material.'
Can I say "I believe I can do this"? Yes, you can.