I'm studying computer science and I realized that I have problems in working with mathematical proofs. They are for example part of my class Formal Systems and Automata. I'm really interested in learning proof theory, but I have no clue where to start and what to learn first.
Could you please give me an advice and also point me to, for example, some literature?
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@goblin has already given you a good list of books that you could have a look at. I second this recommendation.
Another recommendation is to take a math class on something like abstract algebra or discrete mathematics. In studying, for example, abstract algebra you learn how to think abstractly about concepts. You are given definitions of certain things and you are shown proofs of theorems. I think the best way to learn how to do proofs is to practice doing them. So, while abstract algebra is about specific topics like group theory, a good course will be filled with theorems and proofs.
After you have become comfortable working with proofs, you can then ask yourself what specific techniques and tricks you need for your computer science courses.
Note that proof theory isn't the study of how to write an informal proof, its the study of certain logical calculi and the proofs they accept/reject. If I understand correctly, this isn't what you're looking for.
You want something more like:
Good luck.