Reddit mentions of Bridges from Classical to Nonmonotonic Logic (Texts in Computing Vol.5 5)

Priest's introduction to non-classical logic. Goes through all the next steps (modal, intuitionist, fuzzy logics etc.). The formating may take a while to get used to. But a good book.

Edit: It is perfect for finding the logics you find interesting and then you can go more in depth with other books. Also, I think, it is a perfect reference book for logics that I do not know too well (but I do not know if everyone could do this due to have the chapters are arranged and so forth).

Edit2: I would also recommend the following book for nonmonotonic logic, if that floats your boat

In general, the standard turnstile is used to represent a/the consequence relation in standard monotonic logic, whereas the wavy turnstile is used for the consequence relation for non-monotonic logic.

I don't know if this will be at all helpful, but when I was studying non-monotonic logic, I used to think of the wavy turnstile as being similar to the approximately-equals sign - if a = b, then a is always equal to b, but if a ≈ b then there might be some future circumstance in which we cannot consider them to be similar.

In terms of reading material, I can recommend the textbook that we used, D Makinson's Bridges from Classical to Nonmonotonic Logic. I used it as part of a class, but I think Makinson does a great job of explaining concepts and it wouldn't be too bad to go through solo. Also, assuming your professor isn't a total nightmare, I'm sure she will be happy to help you when you get stuck - they love when students make an extra effort to engage with the material.