Suppose I was busy with my exams and now the exams are over. How would I tell a friend that I'm done with my exams. In my language the sentence is something like "I'm free from my exams." What is the correct way of saying this in English?
You already said in your question "I'm done with my exams". That is a much more idiomatic way (in American English at least) to say it than "I'm free from my exams", which doesn't sound quite right. As a British English speaker, I would probably just say "My exams are over", or "I've finished my exams".
"Free from" is used differently, for example, someone might say they are "free from drugs" or "free from blame". Both could mean that the person never had those things. However, saying "I'm free" idiomatically means you are available, so if your intention was to state your availability now your exams have finished, you could say:
I'm free now that my exams have finished.