Scheme theory won't be that important (as you said, most of the time you are concerned with smooth projective varieties over $\mathbb C$). However, you didn't mention about your experience in homological algebra. Huybrechts' book is a pleasure to read, but it can't serve as a textbook in homological algebra. I would recommend e.g. reading first first few chapters of Weibel's book (1, 2, 3, 5, 10). Gelfand-Manin "Methods of Homological Algebra" is also a good reference here.

Moreover, there is a short introduction by R. P. Thomas "Derived categories for the working mathematician".

Scheme theory won't be that important (as you said, most of the time you are concerned with smooth projective varieties over $\mathbb C$). However, you didn't mention about your experience in homological algebra. Huybrechts' book is a pleasure to read, but it can't serve as a textbook in homological algebra. I would recommend e.g. reading first first few chapters of Weibel's book (1, 2, 3, 5, 10). Gelfand-Manin "Methods of Homological Algebra" is also a good reference here.

Moreover, there is a short introduction by R. P. Thomas "Derived categories for the working mathematician".