"나는 공부 많이 하고 있어." I am studying a lot.' is informal "저는 공부 많이 하고 있어요, is formal. -^-^-

"많이 공부했어요" would do, but I think it's kinda... informal? I'm not really sure. If what you really mean is "I've learnt a lot" it would be "많이 배웠어요"